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JMAP REGENTS BY PERFORMANCE INDICATOR: TOPIC NY Algebra 2/Trigonometry Regents Exam Questions from Fall 2009 to August 2015 Sorted by PI: Topic www.jmap.org TABLE OF CONTENTS TOPIC PI: SUBTOPIC QUESTION NUMBER A2.S.1-2: Analysis of Data ............................................................... 1-8 A2.S.3: Average Known with Missing Data ...................................9-10 GRAPHS AND STATISTICS A2.S.4: Dispersion .........................................................................11-19 A2.S.6-7: Regression .....................................................................20-27 A2.S.8: Correlation Coefficient .....................................................28-34 A2.S.5: Normal Distributions ........................................................35-44 A2.S.10: Permutations ...................................................................45-55 A2.S.11: Combinations..................................................................56-61 A2.S.9: Differentiating Permutations and Combinations ..............62-68 PROBABILITY A2.S.12: Sample Space..................................................................69-70 A2.S.13: Geometric Probability.......................................................... 71 A2.S.15: Binomial Probability.......................................................72-83 ABSOLUTE VALUE A2.A.1: Absolute Value Equations and Equalities ........................84-93 A2.A.20-21: Roots of Quadratics ................................................94-106 A2.A.7: Factoring Polynomials .................................................107-109 A2.A.7: Factoring the Difference of Perfect Squares ....................... 110 A2.A.7: Factoring by Grouping .................................................111-115 QUADRATICS A2.A.25: Quadratic Formula .....................................................116-120 A2.A.2: Using the Discriminant ................................................121-128 A2.A.24: Completing the Square...............................................129-135 A2.A.4: Quadratic Inequalities ..................................................136-139 SYSTEMS A2.A.3: Quadratic-Linear Systems ............................................140-144 A2.N.3: Operations with Polynomials .......................................145-153 A2.N.1, A.8-9: Negative and Fractional Exponents ..................154-168 A2.A.12: Evaluating Exponential Expressions ..........................169-175 A2.A.18: Evaluating Logarithmic Expressions .........................176-177 A2.A.53: Graphing Exponential Functions ...............................178-180 POWERS A2.A.54: Graphing Logarithmic Functions ...............................181-183 A2.A.19: Properties of Logarithms............................................184-191 A2.A.28: Logarithmic Equations ...............................................192-203 A2.A.6, 27: Exponential Equations ...........................................204-216 A2.A.36: Binomial Expansions .................................................217-224 A2.A.26, 50: Solving Polynomial Equations.............................225-234 i A2.N.4: Operations with Irrational Expressions ............................... 235 A2.A.13: Simplifying Radicals..................................................236-237 A2.N.2, A.14: Operations with Radicals ...................................238-244 A2.N.5, A.15: Rationalizing Denominators...............................245-253 A2.A.22: Solving Radicals ........................................................254-259 RADICALS A2.A.10-11: Exponents as Radicals ..........................................260-263 A2.N.6: Square Roots of Negative Numbers .............................264-266 A2.N.7: Imaginary Numbers .....................................................267-271 A2.N.8: Conjugates of Complex Numbers ................................272-275 A2.N.9: Multiplication and Division of Complex Numbers ......276-281 A2.A.16: Multiplication and Division of Rationals ...................282-285 A2.A.16: Addition and Subtraction of Rationals .............................. 286 RATIONALS A2.A.23: Solving Rationals and Rational Inequalities ..............287-294 A2.A.17: Complex Fractions .....................................................295-299 A2.A.5: Inverse Variation..........................................................300-307 A2.A.40-41: Functional Notation ..............................................308-311 A2.A.52: Families of Functions........................................................ 312 A2.A.46: Properties of Graphs of Functions and Relations .......313-314 A2.A.52: Identifying the Equation of a Graph...........................315-318 FUNCTIONS A2.A.37, 38, 43: Defining Functions.........................................319-332 A2.A.39, 51: Domain and Range...............................................333-345 A2.A.42: Compositions of Functions ........................................346-354 A2.A.44: Inverse of Functions...................................................355-358 A2.A.46: Transformations with Functions and Relations..........359-362 A2.A.29-33: Sequences .............................................................363-383 SEQUENCES AND SERIES A2.N.10, A.34: Sigma Notation ................................................384-396 A2.A.35: Series..........................................................................397-400 A2.A.55: Trigonometric Ratios .................................................401-406 A2.M.1-2: Radian Measure .......................................................407-420 A2.A.60: Unit Circle .................................................................421-423 A2.A.60: Finding the Terminal Side of an Angle ......................424-425 A2.A.62, 66: Determining Trigonometric Functions.................426-433 TRIGONOMETRY A2.A.64: Using Inverse Trigonometric Functions.....................434-439 A2.A.57: Reference Angles .......................................................440-441 A2.A.61: Arc Length .................................................................442-446 A2.A.58-59: Cofunction/Reciprocal Trigonometric Functions447-456 A2.A.67: Simplifying & Proving Trigonometric Identities .......457-459 A2.A.76: Angle Sum and Difference Identities.........................460-466 ii A2.A.77: Double and Half Angle Identities ..............................467-473 A2.A.68: Trigonometric Equations ...........................................474-483 A2.A.69: Properties of Trigonometric Functions ......................484-489 A2.A.72: Identifying the Equation of a Trigonometric Graph...490-494 A2.A.65, 70-71: Graphing Trigonometric Functions ................495-500 A2.A.63: Domain and Range.....................................................501-503 A2.A.74: Using Trigonometry to Find Area ..............................504-514 A2.A.73: Law of Sines ..............................................................515-518 A2.A.75: Law of Sines - The Ambiguous Case.........................519-526 A2.A.73: Law of Cosines ..........................................................527-533 A2.A.73: Vectors .......................................................................534-536 CONICS A2.A.47-49: Equations of Circles..............................................537-546 iii Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 A school cafeteria has five different lunch periods. The cafeteria staff wants to find out which items on the menu are most popular, so they give every student in the first lunch period a list of questions to answer in order to collect data to represent the school. Which type of study does this represent? 1 observation 2 controlled experiment 3 population survey 4 sample survey GRAPHS AND STATISTICS A2.S.1-2: ANALYSIS OF DATA 1 Howard collected fish eggs from a pond behind his house so he could determine whether sunlight had an effect on how many of the eggs hatched. After he collected the eggs, he divided them into two tanks. He put both tanks outside near the pond, and he covered one of the tanks with a box to block out all sunlight. State whether Howard's investigation was an example of a controlled experiment, an observation, or a survey. Justify your response. 6 A survey completed at a large university asked 2,000 students to estimate the average number of hours they spend studying each week. Every tenth student entering the library was surveyed. The data showed that the mean number of hours that students spend studying was 15.7 per week. Which characteristic of the survey could create a bias in the results? 1 the size of the sample 2 the size of the population 3 the method of analyzing the data 4 the method of choosing the students who were surveyed 2 Which task is not a component of an observational study? 1 The researcher decides who will make up the sample. 2 The researcher analyzes the data received from the sample. 3 The researcher gathers data from the sample, using surveys or taking measurements. 4 The researcher divides the sample into two groups, with one group acting as a control group. 7 The yearbook staff has designed a survey to learn student opinions on how the yearbook could be improved for this year. If they want to distribute this survey to 100 students and obtain the most reliable data, they should survey 1 every third student sent to the office 2 every third student to enter the library 3 every third student to enter the gym for the basketball game 4 every third student arriving at school in the morning 3 A doctor wants to test the effectiveness of a new drug on her patients. She separates her sample of patients into two groups and administers the drug to only one of these groups. She then compares the results. Which type of study best describes this situation? 1 census 2 survey 3 observation 4 controlled experiment 4 A market research firm needs to collect data on viewer preferences for local news programming in Buffalo. Which method of data collection is most appropriate? 1 census 2 survey 3 observation 4 controlled experiment 1 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org A2.S.4: DISPERSION 8 Which survey is least likely to contain bias? 1 surveying a sample of people leaving a movie theater to determine which flavor of ice cream is the most popular 2 surveying the members of a football team to determine the most watched TV sport 3 surveying a sample of people leaving a library to determine the average number of books a person reads in a year 4 surveying a sample of people leaving a gym to determine the average number of hours a person exercises per week 11 The table below shows the first-quarter averages for Mr. Harper’s statistics class. A2.S.3: AVERAGE KNOWN WITH MISSING DATA 9 The number of minutes students took to complete a quiz is summarized in the table below. If the mean number of minutes was 17, which equation could be used to calculate the value of x? 119 + x 1 17 = x 119 + 16x 2 17 = x 446 + x 3 17 = 26 + x 446 + 16x 4 17 = 26 + x What is the population variance for this set of data? 1 8.2 2 8.3 3 67.3 4 69.3 12 The scores of one class on the Unit 2 mathematics test are shown in the table below. 10 The table below displays the results of a survey regarding the number of pets each student in a class has. The average number of pets per student in this class is 2. What is the value of k for this table? 1 9 2 2 3 8 4 4 Find the population standard deviation of these scores, to the nearest tenth. 2 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 13 During a particular month, a local company surveyed all its employees to determine their travel times to work, in minutes. The data for all 15 employees are shown below. 18 The table below shows the final examination scores for Mr. Spear’s class last year. 25 55 40 65 29 45 59 35 25 37 52 30 8 40 55 Determine the number of employees whose travel time is within one standard deviation of the mean. 14 The heights, in inches, of 10 high school varsity basketball players are 78, 79, 79, 72, 75, 71, 74, 74, 83, and 71. Find the interquartile range of this data set. Find the population standard deviation based on these data, to the nearest hundredth. Determine the number of students whose scores are within one population standard deviation of the mean. 15 Ten teams competed in a cheerleading competition at a local high school. Their scores were 29, 28, 39, 37, 45, 40, 41, 38, 37, and 48. How many scores are within one population standard deviation from the mean? For these data, what is the interquartile range? 19 The table below displays the number of siblings of each of the 20 students in a class. 16 The following is a list of the individual points scored by all twelve members of the Webster High School basketball team at a recent game: 2 2 3 4 6 7 9 10 10 11 12 14 Find the interquartile range for this set of data. What is the population standard deviation, to the nearest hundredth, for this group? 1 1.11 2 1.12 3 1.14 4 1.15 17 The table below shows five numbers and their frequency of occurrence. The interquartile range for these data is 1 7 2 5 3 7 to 12 4 6 to 13 3 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 22 The table below shows the number of new stores in a coffee shop chain that opened during the years 1986 through 1994. A2.S.6-7: REGRESSION 20 Samantha constructs the scatter plot below from a set of data. Based on her scatter plot, which regression model would be most appropriate? 1 exponential 2 linear 3 logarithmic 4 power Using x = 1 to represent the year 1986 and y to represent the number of new stores, write the exponential regression equation for these data. Round all values to the nearest thousandth. 21 The table below shows the results of an experiment involving the growth of bacteria. Write a power regression equation for this set of data, rounding all values to three decimal places. Using this equation, predict the bacteria’s growth, to the nearest integer, after 15 minutes. 4 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 23 A population of single-celled organisms was grown in a Petri dish over a period of 16 hours. The number of organisms at a given time is recorded in the table below. 25 The data collected by a biologist showing the growth of a colony of bacteria at the end of each hour are displayed in the table below. Write an exponential regression equation to model these data. Round all values to the nearest thousandth. Assuming this trend continues, use this equation to estimate, to the nearest ten, the number of bacteria in the colony at the end of 7 hours. 26 The table below shows the concentration of ozone in Earth’s atmosphere at different altitudes. Write the exponential regression equation that models these data, rounding all values to the nearest thousandth. Determine the exponential regression equation model for these data, rounding all values to the nearest ten-thousandth. Using this equation, predict the number of single-celled organisms, to the nearest whole number, at the end of the 18th hour. 24 A cup of soup is left on a countertop to cool. The table below gives the temperatures, in degrees Fahrenheit, of the soup recorded over a 10-minute period. 27 The table below shows the amount of a decaying radioactive substance that remained for selected years after 1990. Write an exponential regression equation for the data, rounding all values to the nearest thousandth. Write an exponential regression equation for this set of data, rounding all values to the nearest thousandth. Using this equation, determine the amount of the substance that remained in 2002, to the nearest integer. 5 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org A2.S.8: CORRELATION COEFFICIENT 30 As shown in the table below, a person’s target heart rate during exercise changes as the person gets older. 28 Which value of r represents data with a strong negative linear correlation between two variables? 1 −1.07 2 −0.89 3 −0.14 4 0.92 29 Which calculator output shows the strongest linear relationship between x and y? 1 Which value represents the linear correlation coefficient, rounded to the nearest thousandth, between a person’s age, in years, and that person’s target heart rate, in beats per minute? 1 −0.999 2 −0.664 3 0.998 4 1.503 2 3 31 The relationship between t, a student’s test scores, and d, the student’s success in college, is modeled by the equation d = 0.48t + 75.2. Based on this linear regression model, the correlation coefficient could be 1 between −1 and 0 2 between 0 and 1 3 equal to −1 4 equal to 0 4 32 Which value of r represents data with a strong positive linear correlation between two variables? 1 0.89 2 0.34 3 1.04 4 0.01 6 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 37 Assume that the ages of first-year college students are normally distributed with a mean of 19 years and standard deviation of 1 year. To the nearest integer, find the percentage of first-year college students who are between the ages of 18 years and 20 years, inclusive. To the nearest integer, find the percentage of first-year college students who are 20 years old or older. 33 Determine which set of data given below has the stronger linear relationship between x and y. Justify your choice. 38 In a study of 82 video game players, the researchers found that the ages of these players were normally distributed, with a mean age of 17 years and a standard deviation of 3 years. Determine if there were 15 video game players in this study over the age of 20. Justify your answer. 34 A study compared the number of years of education a person received and that person's average yearly salary. It was determined that the relationship between these two quantities was linear and the correlation coefficient was 0.91. Which conclusion can be made based on the findings of this study? 1 There was a weak relationship. 2 There was a strong relationship. 3 There was no relationship. 4 There was an unpredictable relationship. 39 If the amount of time students work in any given week is normally distributed with a mean of 10 hours per week and a standard deviation of 2 hours, what is the probability a student works between 8 and 11 hours per week? 1 34.1% 2 38.2% 3 53.2% 4 68.2% A2.S.5: NORMAL DISTRIBUTIONS 35 The lengths of 100 pipes have a normal distribution with a mean of 102.4 inches and a standard deviation of 0.2 inch. If one of the pipes measures exactly 102.1 inches, its length lies 1 below the 16th percentile 2 between the 50th and 84th percentiles 3 between the 16th and 50th percentiles 4 above the 84th percentile 40 In a certain high school, a survey revealed the mean amount of bottled water consumed by students each day was 153 bottles with a standard deviation of 22 bottles. Assuming the survey represented a normal distribution, what is the range of the number of bottled waters that approximately 68.2% of the students drink? 1 131 − 164 2 131 − 175 3 142 − 164 4 142 − 175 36 An amateur bowler calculated his bowling average for the season. If the data are normally distributed, about how many of his 50 games were within one standard deviation of the mean? 1 14 2 17 3 34 4 48 41 Liz has applied to a college that requires students to score in the top 6.7% on the mathematics portion of an aptitude test. The scores on the test are approximately normally distributed with a mean score of 576 and a standard deviation of 104. What is the minimum score Liz must earn to meet this requirement? 1 680 2 732 3 740 4 784 7 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 48 A four-digit serial number is to be created from the digits 0 through 9. How many of these serial numbers can be created if 0 can not be the first digit, no digit may be repeated, and the last digit must be 5? 1 448 2 504 3 2,240 4 2,520 42 In a certain school, the heights of the population of girls are normally distributed, with a mean of 63 inches and a standard deviation of 2 inches. If there are 450 girls in the school, determine how many of the girls are shorter than 60 inches. Round the answer to the nearest integer. 43 On a test that has a normal distribution of scores, a score of 57 falls one standard deviation below the mean, and a score of 81 is two standard deviations above the mean. Determine the mean score of this test. 49 How many different six-letter arrangements can be made using the letters of the word “TATTOO”? 1 60 2 90 3 120 4 720 44 The scores on a standardized exam have a mean of 82 and a standard deviation of 3.6. Assuming a normal distribution, a student's score of 91 would rank 1 below the 75th percentile 2 between the 75th and 85th percentiles 3 between the 85th and 95th percentiles 4 above the 95th percentile 50 Find the number of possible different 10-letter arrangements using the letters of the word “STATISTICS.” 51 Which expression represents the total number of different 11-letter arrangements that can be made using the letters in the word “MATHEMATICS”? 11! 1 3! 11! 2 2!+ 2!+ 2! 11! 3 8! 11! 4 2!⋅ 2!⋅ 2! PROBABILITY A2.S.10: PERMUTATIONS 45 Which formula can be used to determine the total number of different eight-letter arrangements that can be formed using the letters in the word DEADLINE? 1 8! 8! 2 4! 8! 3 2!+ 2! 8! 4 2!⋅ 2! 52 The number of possible different 12-letter arrangements of the letters in the word “TRIGONOMETRY” is represented by 12! 1 3! 12! 2 6! 12 P 12 3 8 12 P 12 4 6! 46 The letters of any word can be rearranged. Carol believes that the number of different 9-letter arrangements of the word “TENNESSEE” is greater than the number of different 7-letter arrangements of the word “VERMONT.” Is she correct? Justify your answer. 47 Find the total number of different twelve-letter arrangements that can be formed using the letters in the word PENNSYLVANIA. 8 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 53 How many different 11-letter arrangements are possible using the letters in the word “ARRANGEMENT”? 1 2,494,800 2 4,989,600 3 19,958,400 4 39,916,800 59 If order does not matter, which selection of students would produce the most possible committees? 1 5 out of 15 2 5 out of 25 3 20 out of 25 4 15 out of 25 54 What is the total number of different nine-letter arrangements that can be formed using the letters in the word “TENNESSEE”? 1 3,780 2 15,120 3 45,360 4 362,880 60 How many different ways can teams of four members be formed from a class of 20 students? 1 5 2 80 3 4,845 4 116,280 61 A customer will select three different toppings for a supreme pizza. If there are nine different toppings to choose from, how many different supreme pizzas can be made? 1 12 2 27 3 84 4 504 55 How many distinct ways can the eleven letters in the word "TALLAHASSEE" be arranged? 1 831,600 2 1,663,200 3 3,326,400 4 5,702,400 A2.S.11: COMBINATIONS A2.S.9: DIFFERENTIATING BETWEEN PERMUTATIONS AND COMBINATIONS 56 The principal would like to assemble a committee of 8 students from the 15-member student council. How many different committees can be chosen? 1 120 2 6,435 3 32,432,400 4 259,459,200 62 Twenty different cameras will be assigned to several boxes. Three cameras will be randomly selected and assigned to box A. Which expression can be used to calculate the number of ways that three cameras can be assigned to box A? 1 20! 20! 2 3! 3 20 C 3 4 20 P 3 57 Ms. Bell's mathematics class consists of 4 sophomores, 10 juniors, and 5 seniors. How many different ways can Ms. Bell create a four-member committee of juniors if each junior has an equal chance of being selected? 1 210 2 3,876 3 5,040 4 93,024 58 A blood bank needs twenty people to help with a blood drive. Twenty-five people have volunteered. Find how many different groups of twenty can be formed from the twenty-five volunteers. 9 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 63 Three marbles are to be drawn at random, without replacement, from a bag containing 15 red marbles, 10 blue marbles, and 5 white marbles. Which expression can be used to calculate the probability of drawing 2 red marbles and 1 white marble from the bag? 15 C 2 ⋅5 C 1 1 30 C 3 2 3 4 15 15 15 67 A video-streaming service can choose from six half-hour shows and four one-hour shows. Which expression could be used to calculate the number of different ways the service can choose four half-hour shows and two one-hour shows? 1 6 P 4 ⋅4 P 2 2 6 P4 +4 P2 3 6 C 4 ⋅4 C 2 4 6 C 4 +4 C 2 P 2 ⋅5 P 1 30 C 3 68 Six people met at a dinner party, and each person shook hands once with everyone there. Which expression represents the total number of handshakes? 1 6! 2 6!⋅ 2! 6! 3 2! 6! 4 4!⋅ 2! C 2 ⋅5 C 1 30 P 3 P 2 ⋅5 P 1 30 P 3 64 There are eight people in a tennis club. Which expression can be used to find the number of different ways they can place first, second, and third in a tournament? 1 8 P3 2 8 C3 3 8 P5 4 8 C5 A2.S.12: SAMPLE SPACE 69 A committee of 5 members is to be randomly selected from a group of 9 teachers and 20 students. Determine how many different committees can be formed if 2 members must be teachers and 3 members must be students. 65 Which problem involves evaluating 6 P 4 ? 1 How many different four-digit ID numbers can be formed using 1, 2, 3, 4, 5, and 6 without repetition? 2 How many different subcommittees of four can be chosen from a committee having six members? 3 How many different outfits can be made using six shirts and four pairs of pants? 4 How many different ways can one boy and one girl be selected from a group of four boys and six girls? 70 A school math team consists of three juniors and five seniors. How many different groups can be formed that consist of one junior and two seniors? 1 13 2 15 3 30 4 60 66 A math club has 30 boys and 20 girls. Which expression represents the total number of different 5-member teams, consisting of 3 boys and 2 girls, that can be formed? 1 30 P 3 ⋅20 P 2 2 30 C 3 ⋅20 C 2 3 30 P 3 + 20 P 2 4 30 C 3 + 20 C 2 10 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 73 A study shows that 35% of the fish caught in a local lake had high levels of mercury. Suppose that 10 fish were caught from this lake. Find, to the nearest tenth of a percent, the probability that at least 8 of the 10 fish caught did not contain high levels of mercury. A2.S.13: GEOMETRIC PROBABILITY 71 A dartboard is shown in the diagram below. The two lines intersect at the center of the circle, and 2π . the central angle in sector 2 measures 3 74 The probability that the Stormville Sluggers will 2 win a baseball game is . Determine the 3 probability, to the nearest thousandth, that the Stormville Sluggers will win at least 6 of their next 8 games. 75 The probability that a professional baseball player 1 will get a hit is . Calculate the exact probability 3 that he will get at least 3 hits in 5 attempts. 76 A spinner is divided into eight equal sections. Five sections are red and three are green. If the spinner is spun three times, what is the probability that it lands on red exactly twice? 25 1 64 45 2 512 75 3 512 225 4 512 If darts thrown at this board are equally likely to land anywhere on the board, what is the probability that a dart that hits the board will land in either sector 1 or sector 3? 1 1 6 1 2 3 1 3 2 2 4 3 A2.S.15: BINOMIAL PROBABILITY 72 The members of a men’s club have a choice of wearing black or red vests to their club meetings. A study done over a period of many years determined that the percentage of black vests worn is 60%. If there are 10 men at a club meeting on a given night, what is the probability, to the nearest thousandth, that at least 8 of the vests worn will be black? 11 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 77 A study finds that 80% of the local high school students text while doing homework. Ten students are selected at random from the local high school. Which expression would be part of the process used to determine the probability that, at most, 7 of the 10 students text while doing homework? 4 6 1 4 1 10 C 6 5 5 3 4 ABSOLUTE VALUE A2.A.1: ABSOLUTE VALUE EQUATIONS AND INEQUALITIES 4 1 C 10 7 5 5 7 10 3 2 10 C 8 10 10 10 2 2 . 3 Determine the probability, expressed as a fraction, of winning exactly four games if seven games are played. 83 The probability of winning a game is 7 84 What is the solution set of the equation |4a + 6| − 4a = −10? 1 ∅ 2 0 1 3 2 1 4 0, 2 7 9 3 1 C 10 9 10 10 78 On a multiple-choice test, Abby randomly guesses on all seven questions. Each question has four choices. Find the probability, to the nearest thousandth, that Abby gets exactly three questions correct. 85 What is the solution set of |x − 2| =3x + 10? 1 {} 2 −2 3 {−6} 4 {−2,−6} 79 Because Sam’s backyard gets very little sunlight, the probability that a geranium planted there will flower is 0.28. Sam planted five geraniums. Determine the probability, to the nearest thousandth, that at least four geraniums will flower. 86 Which graph represents the solution set of |6x − 7| ≤ 5? 1 80 Whenever Sara rents a movie, the probability that it is a horror movie is 0.57. Of the next five movies she rents, determine the probability, to the nearest hundredth, that no more than two of these rentals are horror movies. 2 3 4 81 The probability of Ashley being the catcher in a 2 softball game is . Calculate the exact probability 5 that she will be the catcher in exactly five of the next six games. 87 Graph the inequality −3 |6 − x| < −15 for x. Graph the solution on the line below. 82 The probability that Kay and Joseph Dowling will have a redheaded child is 1 out of 4. If the Dowlings plan to have three children, what is the exact probability that only one child will have red hair? 12 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 88 Which graph represents the solution set of | 4x − 5 | | | > 1? | 3 | QUADRATICS A2.A.20-21: ROOTS OF QUADRATICS 94 Find the sum and product of the roots of the equation 5x 2 + 11x − 3 = 0. 1 2 95 What are the sum and product of the roots of the equation 6x 2 − 4x − 12 = 0? 2 1 sum = − ; product = −2 3 2 2 sum = ; product = −2 3 2 3 sum = −2; product = 3 2 4 sum = −2; product = − 3 3 4 89 Determine the solution of the inequality |3 − 2x| ≥ 7. [The use of the grid below is optional.] 96 Determine the sum and the product of the roots of 3x 2 = 11x − 6. 97 Determine the sum and the product of the roots of the equation 12x 2 + x − 6 = 0. 98 What is the product of the roots of the quadratic equation 2x 2 − 7x = 5? 1 5 5 2 2 3 −5 5 4 − 2 90 What is the graph of the solution set of |2x − 1| > 5? 99 What is the product of the roots of 4x 2 − 5x = 3? 3 1 4 5 2 4 3 3 − 4 5 4 − 4 1 2 3 4 91 Solve |−4x + 5| < 13 algebraically for x. 92 Solve |2x − 3| > 5 algebraically. 100 Given the equation 3x 2 + 2x + k = 0, state the sum and product of the roots. 93 Solve algebraically for x: |3x − 5| − x < 17 13 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 101 Which statement about the equation 3x 2 + 9x − 12 = 0 is true? 1 The product of the roots is −12. 2 The product of the roots is −4. 3 The sum of the roots is 3. 4 The sum of the roots is −9. A2.A.7: FACTORING POLYNOMIALS 107 Factored completely, the expression 6x − x 3 − x 2 is equivalent to 1 x(x + 3)(x − 2) 2 x(x − 3)(x + 2) 3 −x(x − 3)(x + 2) 4 −x(x + 3)(x − 2) 102 For which equation does the sum of the roots equal 3 and the product of the roots equal −2? 4 1 4x 2 − 8x + 3 = 0 2 4x 2 + 8x + 3 = 0 3 4x 2 − 3x − 8 = 0 4 4x 2 + 3x − 2 = 0 108 Factored completely, the expression 12x 4 + 10x 3 − 12x 2 is equivalent to 1 x 2 (4x + 6)(3x − 2) 2 3 4 103 For which equation does the sum of the roots equal −3 and the product of the roots equal 2? 1 x 2 + 2x − 3 = 0 2 x 2 − 3x + 2 = 0 3 2x 2 + 6x + 4 = 0 4 2x 2 − 6x + 4 = 0 2(2x 2 + 3x)(3x 2 − 2x) 2x 2 (2x − 3)(3x + 2) 2x 2 (2x + 3)(3x − 2) 109 Factor completely: 10ax 2 − 23ax − 5a A2.A.7: FACTORING THE DIFFERENCE OF PERFECT SQUARES 110 Factor the expression 12t 8 − 75t 4 completely. 104 Write a quadratic equation such that the sum of its roots is 6 and the product of its roots is −27. A2.A.7: FACTORING BY GROUPING 105 Which equation has roots with the sum equal to and the product equal to 1 2 3 4 9 4 111 When factored completely, x 3 + 3x 2 − 4x − 12 equals 1 (x + 2)(x − 2)(x − 3) 2 (x + 2)(x − 2)(x + 3) 3 ? 4 4x 2 + 9x + 3 = 0 4x 2 + 9x − 3 = 0 4x 2 − 9x + 3 = 0 4x 2 − 9x − 3 = 0 3 4 (x 2 − 4)(x + 3) (x 2 − 4)(x − 3) 112 When factored completely, the expression 3x 3 − 5x 2 − 48x + 80 is equivalent to 1 (x 2 − 16)(3x − 5) 106 What is the product of the roots of x 2 − 4x + k = 0 if one of the roots is 7? 1 21 2 −11 3 −21 4 −77 2 3 4 (x 2 + 16)(3x − 5)(3x + 5) (x + 4)(x − 4)(3x − 5) (x + 4)(x − 4)(3x − 5)(3x − 5) 113 The expression x 2 (x + 2) − (x + 2) is equivalent to 1 2 3 4 14 x2 x2 − 1 x 3 + 2x 2 − x + 2 (x + 1)(x − 1)(x + 2) Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 119 A cliff diver on a Caribbean island jumps from a height of 105 feet, with an initial upward velocity of 5 feet per second. An equation that models the height, h(t) , above the water, in feet, of the diver in time elapsed, t, in seconds, is h(t) = −16t 2 + 5t + 105. How many seconds, to the nearest hundredth, does it take the diver to fall 45 feet below his starting point? 1 1.45 2 1.84 3 2.10 4 2.72 114 When factored completely, the expression x 3 − 2x 2 − 9x + 18 is equivalent to 1 (x 2 − 9)(x − 2) 2 (x − 2)(x − 3)(x + 3) 3 4 (x − 2) 2 (x − 3)(x + 3) (x − 3) 2 (x − 2) 115 Factor completely: x 3 − 6x 2 − 25x + 150 A2.A.25: QUADRATIC FORMULA 116 The solutions of the equation y 2 − 3y = 9 are 1 3 ± 3i 3 2 2 3 ± 3i 5 2 3 −3 ± 3 5 2 4 3±3 5 2 120 A homeowner wants to increase the size of a rectangular deck that now measures 14 feet by 22 feet. The building code allows for a deck to have a maximum area of 800 square feet. If the length and width are increased by the same number of feet, find the maximum number of whole feet each dimension can be increased and not exceed the building code. [Only an algebraic solution can receive full credit.] A2.A.2: USING THE DISCRIMINANT 117 The roots of the equation 2x 2 + 7x − 3 = 0 are 1 1 − and −3 2 1 2 and 3 2 3 4 121 Use the discriminant to determine all values of k that would result in the equation x 2 − kx + 4 = 0 having equal roots. −7 ± 73 4 7± 122 The roots of the equation 9x 2 + 3x − 4 = 0 are 1 imaginary 2 real, rational, and equal 3 real, rational, and unequal 4 real, irrational, and unequal 73 4 118 Solve the equation 6x 2 − 2x − 3 = 0 and express the answer in simplest radical form. 123 The roots of the equation x 2 − 10x + 25 = 0 are 1 imaginary 2 real and irrational 3 real, rational, and equal 4 real, rational, and unequal 124 The discriminant of a quadratic equation is 24. The roots are 1 imaginary 2 real, rational, and equal 3 real, rational, and unequal 4 real, irrational, and unequal 15 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 125 The roots of the equation 2x 2 + 4 = 9x are 1 real, rational, and equal 2 real, rational, and unequal 3 real, irrational, and unequal 4 imaginary 131 Brian correctly used a method of completing the square to solve the equation x 2 + 7x − 11 = 0. Brian’s first step was to rewrite the equation as x 2 + 7x = 11 . He then added a number to both sides of the equation. Which number did he add? 7 1 2 49 2 4 49 3 2 4 49 126 For which value of k will the roots of the equation 2x 2 − 5x + k = 0 be real and rational numbers? 1 1 2 −5 3 0 4 4 127 Which equation has real, rational, and unequal roots? 1 x 2 + 10x + 25 = 0 2 x 2 − 5x + 4 = 0 3 x 2 − 3x + 1 = 0 4 x 2 − 2x + 5 = 0 132 Max solves a quadratic equation by completing the square. He shows a correct step: (x + 2) 2 = −9 What are the solutions to his equation? 1 2 ± 3i 2 −2 ± 3i 3 3 ± 2i 4 −3 ± 2i 128 The roots of 3x 2 + x = 14 are 1 imaginary 2 real, rational, and equal 3 real, rational, and unequal 4 real, irrational, and unequal 133 Which step can be used when solving x 2 − 6x − 25 = 0 by completing the square? 1 x 2 − 6x + 9 = 25 + 9 2 x 2 − 6x − 9 = 25 − 9 3 x 2 − 6x + 36 = 25 + 36 4 x 2 − 6x − 36 = 25 − 36 A2.A.24: COMPLETING THE SQUARE 129 Solve 2x 2 − 12x + 4 = 0 by completing the square, expressing the result in simplest radical form. 134 If x 2 = 12x − 7 is solved by completing the square, one of the steps in the process is 1 (x − 6) 2 = −43 130 If x 2 + 2 = 6x is solved by completing the square, an intermediate step would be 1 (x + 3) 2 = 7 2 3 4 2 3 (x − 3) 2 = 7 (x − 3) 2 = 11 (x − 6) 2 = 34 4 16 (x + 6) 2 = −43 (x − 6) 2 = 29 (x + 6) 2 = 29 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 135 Which value of k will make x 2 − A2.A.4: QUADRATIC INEQUALITIES 1 x + k a perfect 4 136 Which graph best represents the inequality y + 6 ≥ x2 − x? square trinomial? 1 1 64 1 2 16 1 3 8 1 4 4 1 2 3 4 17 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 137 The solution set of the inequality x 2 − 3x > 10 is 1 {x | − 2 < x < 5} 2 {x | 0 < x < 3} 3 {x | x < − 2 or x > 5} 4 {x | x < − 5 or x > 2} 143 Which ordered pair is in the solution set of the system of equations shown below? y 2 − x 2 + 32 = 0 3y − x = 0 1 2 3 4 138 Find the solution of the inequality x 2 − 4x > 5, algebraically. 139 What is the solution of the inequality 9 − x 2 < 0? 1 {x | − 3 < x < 3} 2 {x | x > 3 or x < −3} 3 {x | x > 3} 4 {x | x < −3} (2,6) (3,1) (−1,−3) (−6,−2) 144 Determine algebraically the x-coordinate of all points where the graphs of xy = 10 and y = x + 3 intersect. POWERS A2.N.3: OPERATIONS WITH POLYNOMIALS SYSTEMS 2 2 145 Express x − 1 as a trinomial. 3 A2.A.3: QUADRATIC-LINEAR SYSTEMS 140 Which values of x are in the solution set of the following system of equations? y = 3x − 6 3 2 1 x − x − 4 is subtracted from 2 4 5 2 3 x − x + 1, the difference is 2 4 1 1 −x 2 + x − 5 2 1 2 x2 − x + 5 2 2 3 −x − x − 3 4 x2 − x − 3 146 When y = x2 − x − 6 1 2 3 4 0, − 4 0, 4 6, − 2 −6, 2 141 Solve the following systems of equations algebraically: 5 = y − x 4x 2 = −17x + y + 4 147 Express the product of 142 Which ordered pair is a solution of the system of equations shown below? x + y = 5 as a trinomial. (x + 3) 2 + (y − 3) 2 = 53 1 2 3 4 (2,3) (5,0) (−5,10) (−4,9) 18 1 2 1 y − y and 12y + 2 3 3 5 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org x 1 x 1 148 What is the product of − and + ? 4 3 4 3 x2 1 − 1 8 9 2 3 4 3 3 3 2 is x + 1 x − 1 − x − 1 152 The expression 2 2 2 equivalent to 1 0 2 −3x 3 x−2 3 4 4 3x − 2 x2 1 − 16 9 x2 x 1 − − 8 6 9 x2 x 1 − − 16 6 9 1 7 2 3 5 x − x is subtracted from x 2 − x + 2, 4 4 8 8 the difference is 1 1 − x2 − x + 2 4 1 2 x −x+2 2 4 1 1 3 − x2 + x + 2 4 2 1 2 1 x − x−2 4 2 4 153 When 2 3 149 What is the product of x − y 2 and 4 5 2 x + 3 y 2 ? 5 4 1 2 3 4 4 2 9 4 x − y 16 25 9 2 4 x− y 16 25 2 2 3 4 x − y 4 5 4 x 5 A2.N.1, A.8-9: NEGATIVE AND FRACTIONAL EXPONENTS 154 If a = 3 and b = −2, what is the value of the a −2 expression −3 ? b 9 1 − 8 2 −1 8 3 − 9 8 4 9 150 When x 2 + 3x − 4 is subtracted from x 3 + 3x 2 − 2x , the difference is 1 x 3 + 2x 2 − 5x + 4 2 x 3 + 2x 2 + x − 4 3 −x 3 + 4x 2 + x − 4 4 −x 3 − 2x 2 + 5x + 4 2 151 The expression 2 − 3 x is equivalent to 1 4 − 9x 2 4 − 3x 3 4 − 12 x + 9x 4 155 If n is a negative integer, then which statement is always true? 1 6n −2 < 4n −1 n > −6n −1 2 4 3 6n −1 < 4n −1 4 4n −1 > (6n) −1 4 − 12 x + 6x 19 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 1 2 156 What is the value of 4x + x + x 1 1 7 2 1 2 9 2 1 3 16 2 1 4 17 2 0 − 1 4 160 The expression (2a) −4 is equivalent to when x = 16? w −5 157 When simplified, the expression −9 w equivalent to 1 w −7 2 w2 3 w7 4 w 14 1 2 3 4 161 The expression 1 2 is 1 2 3 4 1 3 4 a 2 b −3 is equivalent to a −4 b 2 a6 b5 b5 a6 a2 b a −2 b −1 162 When x −1 − 1 is divided by x − 1, the quotient is 1 −1 1 2 − x 1 3 x2 1 4 (x − 1) 2 − 158 Which expression is equivalent to 9x 2 y 6 2 ? 1 1 3xy 3 2 −8a 4 16 a4 2 − 4 a 1 16a 4 3xy 3 3 xy 3 xy 3 3 163 Simplify the expression 3x −4 y 5 and write the (2x 3 y −7 ) −2 answer using only positive exponents. −1 159 Which expression is equivalent to 3x 2 ? 1 1 3x 2 2 −3x 2 1 3 9x 2 4 −9x 2 164 When x −1 + 1 is divided by x + 1, the quotient equals 1 1 1 2 x 3 x 1 4 − x 20 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 165 Which expression is equivalent to 1 2 3 4 x −1 y 4 3x −5 y −1 y 3x 5 3 4 y 1 x y2 2 x3 y6 3 4 4 166 Which expression is equivalent to 2 168 Which expression is equivalent to x4 y5 3 5 4 x y 3 3x 4 y 5 −2 1 ? 2x y 4y −5 −2 x −1 y 2 x 2 y −4 ? y2 x y6 x3 ? A2.A.12: EVALUATING EXPONENTIAL EXPRESSIONS 3 2x 2 2y 3 169 Matt places $1,200 in an investment account earning an annual rate of 6.5%, compounded continuously. Using the formula V = Pe rt , where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, that Matt will have in the account after 10 years. x2 2x 2 y3 x2 2y 3 167 Which expression is equivalent to (5 −2 a 3 b −4 ) −1 ? 170 Evaluate e x ln y when x = 3 and y = 2 . 4 1 2 3 4 10b a3 25b 4 a3 a3 25b 4 a2 125b 5 171 The formula for continuously compounded interest is A = Pe rt , where A is the amount of money in the account, P is the initial investment, r is the interest rate, and t is the time in years. Using the formula, determine, to the nearest dollar, the amount in the account after 8 years if $750 is invested at an annual rate of 3%. 21 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 175 Yusef deposits $50 into a savings account that pays 3.25% interest compounded quarterly. The amount, A, in his account can be determined by the nt r formula A = P 1 + , where P is the initial n 172 If $5000 is invested at a rate of 3% interest compounded quarterly, what is the value of the investment in 5 years? (Use the formula nt r A = P 1 + , where A is the amount accrued, P n is the principal, r is the interest rate, n is the number of times per year the money is compounded, and t is the length of time, in years.) 1 $5190.33 2 $5796.37 3 $5805.92 4 $5808.08 amount invested, r is the interest rate, n is the number of times per year the money is compounded, and t is the number of years for which the money is invested. What will his investment be worth in 12 years if he makes no other deposits or withdrawals? 1 $55.10 2 $73.73 3 $232.11 4 $619.74 173 The formula to determine continuously compounded interest is A = Pe rt , where A is the amount of money in the account, P is the initial investment, r is the interest rate, and t is the time, in years. Which equation could be used to determine the value of an account with an $18,000 initial investment, at an interest rate of 1.25% for 24 months? 1 A = 18,000e 1.25 • 2 2 A = 18,000e 1.25 • 24 3 A = 18,000e 0.0125 • 2 4 A = 18,000e 0.0125 • 24 A2.A.18: EVALUATING LOGARITHMIC EXPRESSIONS 176 The expression log 8 64 is equivalent to 1 8 2 2 1 3 2 1 4 8 174 A population, p(x) , of wild turkeys in a certain area 1 177 The expression log 5 is equivalent to 25 1 1 2 2 2 1 3 − 2 4 −2 is represented by the function p(x) = 17(1.15) 2x , where x is the number of years since 2010. How many more turkeys will be in the population for the year 2015 than 2010? 1 46 2 49 3 51 4 68 22 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 179 On the axes below, for −2 ≤ x ≤ 2, graph y = 2x + 1 − 3. A2.A.53: GRAPHING EXPONENTIAL FUNCTIONS x 1 178 The graph of the equation y = has an 2 asymptote. On the grid below, sketch the graph of x 1 y = and write the equation of this asymptote. 2 23 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 180 An investment is earning 5% interest compounded quarterly. The equation represents the total amount of money, A, where P is the original investment, r is the interest rate, t is the number of years, and n represents the number of times per year the money earns interest. Which graph could represent this investment over at least 50 years? A2.A.54: GRAPHING LOGARITHMIC FUNCTIONS 181 If a function is defined by the equation f(x) = 4 x , which graph represents the inverse of this function? 1 1 2 2 3 3 4 4 24 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 182 Which graph represents the function log 2 x = y ? 183 Which sketch shows the inverse of y = a x , where a > 1? 1 1 2 2 3 3 4 A2.A.19: PROPERTIES OF LOGARITHMS 184 The expression 2logx − (3log y + log z) is equivalent to x2 1 log 3 y z 4 2 3 4 25 x2z y3 2x log 3yz 2xz log 3y log Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 185 If 1 2 3 4 189 The expression log4m2 is equivalent to 1 2(log4 + logm) 2 2log4 + log m 3 log4 + 2logm 4 log16 + 2logm A2 B r= , then log r can be represented by C 1 1 log A + log B − log C 3 6 2 3(log A + log B − logC) 1 log(A 2 + B) − C 3 1 1 2 log A + log B − log C 3 3 3 3 190 If 2x 3 = y, then logy equals 1 log(2x) + log3 2 3log(2x) 3 3log2 + 3 logx 4 log2 + 3logx 186 If log x 2 − log 2a = log3a , then logx expressed in terms of loga is equivalent to 1 log 5a 1 2 1 log 6 + log a 2 2 3 log6 + loga 4 log6 + 2loga 191 If logx = 2log a + log b , then x equals 1 2 3 4 a2b 2ab a2 + b 2a + b A2.A.28: LOGARITHMIC EQUATIONS 1 187 If log b x = 3log b p − 2 log b t + log b r , then the 2 value of x is p3 1 t2 r 2 3 4 3 2 p t r 192 What is the solution of the equation 2 log 4 (5x) = 3? 1 6.4 2 2.56 9 3 5 8 4 5 1 2 p 3 t2 193 Solve algebraically for x: log x + 3 r x3 + x − 2 =2 x p3 t2 194 The temperature, T, of a given cup of hot chocolate after it has been cooling for t minutes can best be modeled by the function below, where T 0 is the temperature of the room and k is a constant. ln(T − T 0 ) = −kt + 4.718 A cup of hot chocolate is placed in a room that has a temperature of 68°. After 3 minutes, the temperature of the hot chocolate is 150°. Compute the value of k to the nearest thousandth. [Only an algebraic solution can receive full credit.] Using this value of k, find the temperature, T, of this cup of hot chocolate if it has been sitting in this room for a total of 10 minutes. Express your answer to the nearest degree. [Only an algebraic solution can receive full credit.] r 188 If log2 = a and log3 = b , the expression log 9 is 20 equivalent to 1 2b − a + 1 2 2b − a − 1 3 b 2 − a + 10 2b 4 a+1 26 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 195 What is the value of x in the equation log 5 x = 4? 1 1.16 2 20 3 625 4 1,024 A2.A.6, 27: EXPONENTIAL EQUATIONS 204 Akeem invests $25,000 in an account that pays 4.75% annual interest compounded continuously. Using the formula A = Pe rt , where A = the amount in the account after t years, P = principal invested, and r = the annual interest rate, how many years, to the nearest tenth, will it take for Akeem’s investment to triple? 1 10.0 2 14.6 3 23.1 4 24.0 3 196 If log 4 x = 2.5 and log y 125 = − , find the numerical 2 x value of , in simplest form. y 197 Solve algebraically for all values of x: log (x + 4) (17x − 4) = 2 205 A population of rabbits doubles every 60 days 4 198 Solve algebraically for x: log 27 (2x − 1) = 3 t 60 according to the formula P = 10(2) , where P is the population of rabbits on day t. What is the value of t when the population is 320? 1 240 2 300 3 660 4 960 199 Solve algebraically for all values of x: log (x + 3) (2x + 3) + log (x + 3) (x + 5) = 2 200 Solve algebraically for x: log5x − 1 4 = 1 3 206 The number of bacteria present in a Petri dish can be modeled by the function N = 50e 3t , where N is the number of bacteria present in the Petri dish after t hours. Using this model, determine, to the nearest hundredth, the number of hours it will take for N to reach 30,700. 201 The equation log a x = y where x > 0 and a > 1 is equivalent to y 1 2 x =a ya = x 3 4 a =x ax = y y 207 Susie invests $500 in an account that is compounded continuously at an annual interest rate of 5%, according to the formula A = Pe rt , where A is the amount accrued, P is the principal, r is the rate of interest, and t is the time, in years. Approximately how many years will it take for Susie’s money to double? 1 1.4 2 6.0 3 13.9 4 14.7 202 If log x + 1 64 = 3, find the value of x. 203 Solve algebraically, to the nearest hundredth, for all values of x: log 2 (x 2 − 7x + 12) − log 2 (2x − 10) = 3 27 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 208 The solution set of 4 x 1 {1,3} 2 {−1,3} 3 {−1,−3} 4 {1,−3} 2 + 4x A2.A.36: BINOMIAL EXPANSIONS = 2 −6 is 217 What is the fourth term in the expansion of (3x − 2) 5 ? 1 2 3 4 209 What is the value of x in the equation 9 3x + 1 = 27 x + 2 ? 1 1 1 2 3 1 3 2 4 4 3 218 Write the binomial expansion of (2x − 1) 5 as a polynomial in simplest form. 219 What is the coefficient of the fourth term in the expansion of (a − 4b) 9 ? 1 −5,376 2 −336 3 336 4 5,376 210 Solve algebraically for x: 16 2x + 3 = 64 x + 2 211 The value of x in the equation 4 2x + 5 = 8 3x is 1 1 2 2 3 5 4 −10 220 Which expression represents the third term in the expansion of (2x 4 − y) 3 ? 1 2 212 Solve algebraically for all values of x: 81 x 3 + 2x 2 = 27 3 5x 3 4 −y 3 −6x 4 y 2 6x 4 y 2 2x 4 y 2 221 What is the middle term in the expansion of x 6 − 2y ? 2 213 Which value of k satisfies the equation 8 3k + 4 = 4 2k − 1 ? 1 −1 9 2 − 4 3 −2 14 4 − 5 1 2 3 4 214 Solve e = 12 algebraically for x, rounded to the nearest hundredth. 4x 20x 3 y 3 15 − x4y2 4 −20x 3 y 3 15 4 2 x y 4 222 What is the fourth term in the binomial expansion (x − 2) 8 ? 215 Solve algebraically for x: 5 4x = 125 x − 1 216 Solve for x: −720x 2 −240x 720x 2 1,080x 3 1 2 3 4 1 = 2 3x − 1 16 28 448x 5 448x 4 −448x 5 −448x 4 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 231 The graph of y = f(x) is shown below. 223 What is the third term in the expansion of (2x − 3) 5 ? 1 2 3 4 720x 3 180x 3 −540x 2 −1080x 2 224 The ninth term of the expansion of (3x + 2y) 15 is 1 15 C 9 (3x) 6 (2y) 9 2 15 C 9 (3x) 9 (2y) 6 3 15 C 8 (3x) 7 (2y) 8 4 15 C 8 (3x) 8 (2y) 7 A2.A.26, 50: SOLVING POLYNOMIAL EQUATIONS 225 Solve the equation 8x 3 + 4x 2 − 18x − 9 = 0 algebraically for all values of x. Which set lists all the real solutions of f(x) = 0? 1 {−3,2} 2 {−2,3} 3 {−3,0,2} 4 {−2,0,3} 226 Which values of x are solutions of the equation x 3 + x 2 − 2x = 0? 1 0,1,2 2 0,1,−2 3 0,−1,2 4 0,−1,−2 227 What is the solution set of the equation 3x 5 − 48x = 0? 1 {0,±2} 2 {0,±2,3} 3 {0,±2,±2i} 4 {±2,±2i} 228 Solve algebraically for all values of x: x 4 + 4x 3 + 4x 2 = −16x 229 Solve x 3 + 5x 2 = 4x + 20 algebraically. 230 Solve the equation 2x 3 − x 2 − 8x + 4 = 0 algebraically for all values of x. 29 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 232 The graph of y = x 3 − 4x 2 + x + 6 is shown below. 234 What are the zeros of the polynomial function graphed below? 1 2 3 4 What is the product of the roots of the equation x 3 − 4x 2 + x + 6 = 0? 1 −36 2 −6 3 6 4 4 {−3,−1,2} {3,1,−2} {4,−8} {−6} RADICALS A2.N.4: OPERATIONS WITH IRRATIONAL EXPRESSIONS 235 The product of (3 + 233 How many negative solutions to the equation 2x 3 − 4x 2 + 3x − 1 = 0 exist? 1 1 2 2 3 3 4 0 1 2 3 4 5) and (3 − 5) is 4−6 5 14 − 6 5 14 4 A2.A.13: SIMPLIFYING RADICALS 236 Express in simplest form: 237 The expression 1 8a 4 2 8a 8 3 4a 5 3 a 4 30 4a 3 a5 3 3 a6b9 −64 64a 16 is equivalent to Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org A2.N.2, A.14: OPERATIONS WITH RADICALS 244 The expression 1 6ab 2 2 6ab 4 3 12ab 2 4 12ab 4 238 Express 5 3x 3 − 2 27x 3 in simplest radical form. 3 239 The sum of 6a 4 b 2 and simplest radical form, is 1 6 2 2a 2 b 3 4 3 162a 4 b 2 , expressed in 21a 2 b 3 3 3 240 The expression 27x 2 16x 4 is equivalent to 1 12x 2 3 2 3 6x 2x 2 6x 2 3 2 4 2 3 4 8ab 2 a 2 8a 2 b 3 3 a 4 4a 2 b 4 and 3 1 − 14 + 5 3 11 2 − 17 + 5 3 11 3 14 + 5 3 14 4 17 + 5 3 14 16a 3 b 2 ? 3 247 The expression 2b − 3a 242 The expression 4ab is equivalent to 1 2ab 6b 3 3 3 4ab 2 a 2 4a 2 b 3 3 a 2 246 Which expression is equivalent to 3 241 What is the product of 1 18b + 7ab 3 1 6b 2 16ab 2b −5ab + 7ab 6b −5ab 2b + 7ab 3 6b 4 243 Express 108x 5 y 8 16b 8 is equivalent to with a rational denominator, in 3− 2 simplest radical form. 4a 6ab 10a 2 b 3 8 12x 3 2x 4 5 245 Express 2 2 27a 3 ⋅ A2.N.5, A.15: RATIONALIZING DENOMINATORS 168a 8 b 4 3 3 in simplest radical form. 6xy 5 31 4 5− 4 13 5 13 − 13 4(5 − 13) 38 5+ 13 3 4(5 + 13) 38 13 3 +5 3 −5 is equivalent to ? Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic 1 248 The expression 1 2 3 4 7+ 7− 2 3 4 7− 1 11 38 7+ 60 4+ 2 3 4 4− 11 is equivalent to 3b ab 4 3 a 252 The expression 1 20 + 5 11 27 4 − 11 2 20 − 5 11 27 3 4 8 3 3 3− 1 2 3 6 2 24 3 3− 2 3 2x + 4 is equivalent to x+2 (2x + 4) x − 2 x−2 (2x + 4) x − 2 x−4 2 x−2 2 x+2 253 Expressed with a rational denominator and in x is simplest form, x− x is equivalent to 3 −2 6 − 3+ b 3 11 3− a b ab 11 5 is equivalent to 1 2 11 60 7− 3 3a 2 b 38 250 The expression 1 251 The fraction 11 249 The expression 1 11 is equivalent to 6 x2 + x x x2 − x − x 3 x+ x 1−x 4 x+ x x−1 A2.A.22: SOLVING RADICALS 254 The solution set of the equation 1 {1} 2 {0} 3 {1,6} 4 {2,3} 32 x + 3 = 3 − x is Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 255 The solution set of 1 {−3,4} 2 {−4,3} 3 {3} 4 {−4} 3x + 16 = x + 2 is − 261 The expression x 256 Solve algebraically for x: 4 − 1 − x5 2 − x2 1 3 2x − 5 = 1 4 257 What is the solution set for the equation 5x + 29 = x + 3? 1 {4} 2 {−5} 3 {4,5} 4 {−5,4} 2 5 2 x5 1 5 x2 3 4 259 The solution set of the equation 1 {−2,−4} 2 {2,4} 3 {4} 4 { } 1 2 7 4 1 2 7 4 263 The expression 1 3 260 The expression (x 2 − 1) 1 3 2 3 3 4 (x − 1) 1 2 4 81x 2 y 5 is equivalent to 2x − 4 = x − 2 is 2 2 3 16x 2 y 7 is equivalent to 4x y 4x 8 y 28 1 2 5 4 1 2 4 5 3x y 3x y A2.A.10-11: EXPONENTS AS RADICALS − 4 2x y 2x 8 y 28 258 Solve algebraically for x: x 2 + x − 1 + 11x = 7x + 3 is equivalent to 2 262 The expression 1 2 5 is equivalent to 4 9xy 9xy 5 2 2 5 2 A2.N.6: SQUARE ROOTS OF NEGATIVE NUMBERS (x 2 − 1) 2 264 In simplest form, (x 2 − 1) 3 1 1 2 (x 2 − 1) 3 3 4 3i 5i 10i 12i −300 is equivalent to 10 12 3 5 265 Expressed in simplest form, 1 2 3 4 33 − −7 −i 7i 2 2 2 2 −18 − −32 is Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org −180x 16 is equivalent to 266 The expression 1 2 3 4 274 What is the conjugate of −6x 4 5 −6x 8 5 6x 4 i 5 6x 8 i 5 1 2 3 A2.N.7: IMAGINARY NUMBERS 4 7 5 267 The product of i and i is equivalent to 1 1 2 −1 3 i 4 −i 1 3 − + i 2 2 1 3 − i 2 2 3 1 + i 2 2 1 3 − − i 2 2 275 The conjugate of the complex expression −5x + 4i is 1 5x − 4i 2 5x + 4i 3 −5x − 4i 4 −5x + 4i 268 The expression 2i 2 + 3i 3 is equivalent to 1 −2 − 3i 2 2 − 3i 3 −2 + 3i 4 2 + 3i A2.N.9: MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS 276 The expression (3 − 7i) 2 is equivalent to 1 −40 + 0i 2 −40 − 42i 3 58 + 0i 4 58 − 42i 269 Determine the value of n in simplest form: i 13 + i 18 + i 31 + n = 0 270 Express 4xi + 5yi 8 + 6xi 3 + 2yi 4 in simplest a + bi form. 277 The expression (x + i) 2 − (x − i) 2 is equivalent to 1 0 2 −2 3 −2 + 4xi 4 4xi 271 Express xi − yi in simplest form. 8 1 3 + i? 2 2 6 A2.N.8: CONJUGATES OF COMPLEX NUMBERS 278 If x = 3i , y = 2i , and z = m + i, the expression xy 2 z equals 1 −12 − 12mi 2 −6 − 6mi 3 12 − 12mi 4 6 − 6mi 272 What is the conjugate of −2 + 3i? 1 −3 + 2i 2 −2 − 3i 3 2 − 3i 4 3 + 2i 273 The conjugate of 7 − 5i is 1 −7 − 5i 2 −7 + 5i 3 7 − 5i 4 7 + 5i 279 Multiply x + yi by its conjugate, and express the product in simplest form. 34 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 280 When −3 − 2i is multiplied by its conjugate, the result is 1 −13 2 −5 3 5 4 13 A2.A.16: ADDITION AND SUBTRACTION OF RATIONALS 286 Expressed in simplest form, equivalent to −6y 2 + 36y − 54 1 (2y − 6)(6 − 2y) 3y − 9 2 2y − 6 3 3 2 3 4 − 2 281 If x is a real number, express 2xi(i − 4i 2 ) in simplest a + bi form. RATIONALS A2.A.16: MULTIPLICATION AND DIVISION OF RATIONALS 282 Perform the indicated operations and simplify completely: x 3 − 3x 2 + 6x − 18 2x − 4 x 2 + 2x − 8 ⋅ ÷ x 2 − 4x 16 − x 2 x 4 − 3x 3 A2.A.23: SOLVING RATIONALS AND RATIONAL INEQALITIES 287 Solve for x: 4 − x2 x 2 + 7x + 12 283 Express in simplest form: 2x − 4 x+3 284 The expression to 1 2 3 4 3y 9 + is 2y − 6 6 − 2y 12 4x = 2+ x−3 x−3 288 Solve algebraically for x: x 2 + 9x − 22 ÷ (2 − x) is equivalent x 2 − 121 2 4 1 − = x + 3 3 − x x2 − 9 289 Solve the equation below algebraically, and express the result in simplest radical form: 13 = 10 − x x x − 11 1 x − 11 11 − x 1 11 − x 290 What is the solution set of the equation 5 30 +1= ? 2 x − 3 x −9 1 {2,3} 2 {2} 3 {3} 4 {} 36 − x 2 (x + 6) 2 285 Express in simplest form: x−3 2 x + 3x − 18 291 Which equation could be used to solve 2 5 − = 1? x−3 x 1 x 2 − 6x − 3 = 0 2 x 2 − 6x + 3 = 0 3 x 2 − 6x − 6 = 0 4 x 2 − 6x + 6 = 0 35 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 292 Solve algebraically for x: x 2 3 + = − x+2 x x+2 1− 297 The simplest form of 293 Solve algebraically for the exact values of x: 5x 1 x = + x 4 2 1 2 294 Which graph represents the solution set of x + 16 ≤ 7? x−2 3 1 4 2 1− 4 x 2 8 − x x2 is 1 2 x x+2 x 3 x − x−2 b c 298 The expression is equivalent to b d− c c+1 1 d−1 a +b 2 d −b ac + b 3 cd − b ac + 1 4 cd − 1 a+ 3 4 A2.A.17: COMPLEX FRACTIONS x 1 − 4 x 295 Written in simplest form, the expression is 1 1 + 2x 4 equivalent to 1 x−1 2 x−2 x−2 3 2 2 x −4 4 x+2 1+ 299 Express in simplest terms: 1 4 − 2 d 296 Express in simplest form: 3 1 + d 2d 1− 3 x 5 24 − x x2 A2.A.5: INVERSE VARIATION 300 For a given set of rectangles, the length is inversely proportional to the width. In one of these rectangles, the length is 12 and the width is 6. For this set of rectangles, calculate the width of a rectangle whose length is 9. 36 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 307 A scholarship committee rewards the school's top math students. The amount of money each winner receives is inversely proportional to the number of scholarship recipients. If there are three winners, they each receive $400. If there are eight winners, how much money will each winner receive? 1 $1067 2 $400 3 $240 4 $150 3 301 If p varies inversely as q, and p = 10 when q = , 2 3 what is the value of p when q = ? 5 1 25 2 15 3 9 4 4 302 The quantities p and q vary inversely. If p = 20 when q = −2, and p = x when q = −2x + 2, then x equals 1 −4 and 5 20 2 19 3 −5 and 4 1 4 − 4 303 The points (2,3), FUNCTIONS A2.A.40-41: FUNCTIONAL NOTATION 308 The equation y − 2 sin θ = 3 may be rewritten as 1 f(y) = 2 sinx + 3 2 f(y) = 2sin θ + 3 3 f(x) = 2sin θ + 3 4 f(θ) = 2sin θ + 3 3 4, , and (6,d) lie on the graph 4 309 If f x = of a function. If y is inversely proportional to the square of x, what is the value of d? 1 1 1 2 3 3 3 4 27 1 2 3 304 If d varies inversely as t, and d = 20 when t = 2, what is the value of t when d = −5? 1 8 2 2 3 −8 4 −2 4 x , what is the value of f(−10)? x − 16 2 5 2 5 − 42 5 58 5 18 − 310 If g(x) = ax form. 305 If p and q vary inversely and p is 25 when q is 6, determine q when p is equal to 30. 2 1 − x , express g(10) in simplest 311 If f x = 4x 2 − x + 1, then f(a + 1) equals 1 2 3 4 306 Given y varies inversely as x, when y is multiplied 1 by , then x is multiplied by 2 1 1 2 2 2 1 3 − 2 4 −2 37 4a 2 − a + 6 4a 2 − a + 4 4a 2 + 7a + 6 4a 2 + 7a + 4 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org A2.A.52: FAMILIES OF FUNCTIONS A2.A.52: PROPERTIES OF GRAPHS OF FUNCTIONS AND RELATIONS 312 On January 1, a share of a certain stock cost $180. Each month thereafter, the cost of a share of this stock decreased by one-third. If x represents the time, in months, and y represents the cost of the stock, in dollars, which graph best represents the cost of a share over the following 5 months? 313 Which statement about the graph of the equation y = e x is not true? 1 It is asymptotic to the x-axis. 2 The domain is the set of all real numbers. 3 It lies in Quadrants I and II. 4 It passes through the point (e,1). 314 Theresa is comparing the graphs of y = 2 x and y = 5 x . Which statement is true? 1 The y-intercept of y = 2 x is (0,2), and the y-intercept of y = 5 x is (0,5). 2 Both graphs have a y-intercept of (0,1), and y = 2 x is steeper for x > 0 . 3 Both graphs have a y-intercept of (0,1), and y = 5 x is steeper for x > 0 . 4 Neither graph has a y-intercept. 1 A2.A.52: IDENTIFYING THE EQUATION OF A GRAPH 2 315 Four points on the graph of the function f(x) are shown below. {(0,1),(1,2),(2,4),(3,8)} Which equation represents f(x)? 1 2 3 4 3 4 38 f(x) = 2 x f(x) = 2x f(x) = x + 1 f(x) = log 2 x Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 316 Which equation is represented by the graph below? 318 The table of values below can be modeled by which equation? 1 2 3 4 1 2 3 4 y = 5x y = 0.5 x y = 5 −x y = 0.5−x 319 Given the relation {(8,2),(3,6),(7,5),(k,4)}, which value of k will result in the relation not being a function? 1 1 2 2 3 3 4 4 below? 2 y = 2x y = 2 −x 3 x=2 y 4 x=2 −y |x + 3 | |x | + 3 | y + 3| |y| + 3 A2.A.37, 38, 43: DEFINING FUNCTIONS 317 What is the equation of the graph shown 1 f(x) = f(x) = f(y) = f(y) = 39 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 320 Which graph does not represent a function? 322 Which graph does not represent a function? 1 1 2 2 3 3 4 4 321 Which relation is not a function? 1 (x − 2) 2 + y 2 = 4 2 3 4 x 2 + 4x + y = 4 x+y = 4 xy = 4 40 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 323 Which graph represents a relation that is not a function? 324 Which graph represents a function? 1 1 2 2 3 3 4 4 41 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 325 Which statement is true about the graphs of f and g shown below? 327 Which graph represents a one-to-one function? 1 2 1 2 3 4 f is a relation and g is a function. f is a function and g is a relation. Both f and g are functions. Neither f nor g is a function. 3 326 Which function is not one-to-one? 1 {(0,1),(1,2),(2,3),(3,4)} 2 {(0,0),(1,1),(2,2),(3,3)} 3 {(0,1),(1,0),(2,3),(3,2)} 4 {(0,1),(1,0),(2,0),(3,2)} 4 328 Which function is one-to-one? 1 f(x) = |x | 2 3 4 f(x) = 2 x f(x) = x 2 f(x) = sinx 329 Which function is one-to-one? 1 k(x) = x 2 + 2 2 3 4 42 g(x) = x 3 + 2 f(x) = |x | + 2 j(x) = x 4 + 2 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 330 Which diagram represents a relation that is both one-to-one and onto? 332 Which list of ordered pairs does not represent a one-to-one function? 1 (1,−1),(2,0),(3,1),(4,2) 2 (1,2),(2,3),(3,4),(4,6) 3 (1,3),(2,4),(3,3),(4,1) 4 (1,5),(2,4),(3,1),(4,0) 1 A2.A.39, 51: DOMAIN AND RANGE 333 What is the domain of the function f(x) = x − 2 + 3? 1 (−∞,∞) 2 (2,∞) 3 [2,∞) 4 [3,∞) 2 3 334 What is the range of f(x) = (x + 4) 2 + 7? 1 y ≥ −4 2 y≥4 3 y=7 4 y≥7 4 331 Which relation is both one-to-one and onto? 335 What is the range of f(x) = |x − 3 | + 2? 1 {x | x ≥ 3} 2 {y | y ≥ 2} 3 {x | x ∈ real numbers} 4 {y | y ∈ real numbers} 1 336 If f(x) = 9 − x 2 , what are its domain and range? 1 domain: {x | − 3 ≤ x ≤ 3}; range: {y | 0 ≤ y ≤ 3} 2 domain: {x | x ≠ ±3}; range: {y | 0 ≤ y ≤ 3} 3 domain: {x | x ≤ −3 or x ≥ 3}; range: {y | y ≠ 0} 4 domain: {x | x ≠ 3}; range: {y | y ≥ 0} 2 3 3 , what are the domain and range? x−4 {x | x > 4} and {y | y > 0} {x | x ≥ 4} and {y | y > 0} {x | x > 4} and {y | y ≥ 0} {x | x ≥ 4} and {y | y ≥ 0} 337 For y = 1 2 3 4 4 43 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 338 The domain of f(x) = − 341 The graph below represents the function y = f(x) . 3 is the set of all real 2−x numbers 1 greater than 2 2 less than 2 3 except 2 4 between −2 and 2 339 What is the domain of the function g x = 3 x -1? 1 (−∞,3] 2 (−∞,3) 3 (−∞,∞) 4 (−1,∞) 340 What are the domain and the range of the function shown in the graph below? State the domain and range of this function. 342 What is the domain of the function shown below? 1 2 3 4 {x | x > −4}; {y | y > 2} {x | x ≥ −4}; {y | y ≥ 2} {x | x > 2}; {y | y > −4} {x | x ≥ 2}; {y | y ≥ −4} 1 2 3 4 44 −1 ≤ x ≤ 6 −1 ≤ y ≤ 6 −2 ≤ x ≤ 5 −2 ≤ y ≤ 5 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 343 What is the range of the function shown below? 1 2 3 4 345 Which value is in the domain of the function graphed below, but is not in its range? x≤0 x≥0 y≤0 y≥0 1 2 3 4 344 The graph below shows the average price of gasoline, in dollars, for the years 1997 to 2007. 0 2 3 7 A2.A.42: COMPOSITIONS OF FUNCTIONS 1 x − 3 and g(x) = 2x + 5, what is the value 2 of (g f)(4)? 1 −13 2 3.5 3 3 4 6 346 If f(x) = 347 If f(x) = x 2 − 5 and g(x) = 6x, then g(f(x)) is equal to 1 6x 3 − 30x 2 6x 2 − 30 3 36x 2 − 5 4 x 2 + 6x − 5 What is the approximate range of this graph? 1 1997 ≤ x ≤ 2007 2 1999 ≤ x ≤ 2007 3 0.97 ≤ y ≤ 2.38 4 1.27 ≤ y ≤ 2.38 348 If f(x) = x 2 − 6 and g(x) = 2 x − 1, determine the value of (g f)(−3). 45 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 349 If f(x) = 4x − x 2 and g(x) = 1 1 , then (f g) is x 2 A2.A.44: INVERSE OF FUNCTIONS 355 Which two functions are inverse functions of each other? 1 f(x) = sinx and g(x) = cos(x) 2 f(x) = 3 + 8x and g(x) = 3 − 8x equal to 4 1 7 2 −2 7 3 2 4 4 given m(x) = sinx , n(x) = 3x , and p(x) = x 2 ? 2 3 4 4 1 f(x) = 2x − 4 and g(x) = − x + 4 2 357 What is the inverse of the function f(x) = log 4 x? sin(3x) 2 3sinx 2 sin 2 (3x) 3sin 2 x 3 f −1 (x) = x 4 f −1 (x) = 4 x f −1 (x) = log x 4 4 f −1 (x) = −log x 4 1 2 1 1 351 If g(x) = x + 8 and h(x) = x − 2, what is the 2 2 value of g(h( − 8))? 1 0 2 9 3 5 4 4 358 If m = {(−1,1),(1,1),(−2,4),(2,4),(−3,9),(3,9)}, which statement is true? 1 m and its inverse are both functions. 2 m is a function and its inverse is not a function. 3 m is not a function and its inverse is a function. 4 Neither m nor its inverse is a function. 352 If f(x) = 2x 2 − 3x + 1 and g(x) = x + 5, what is f(g(x))? 1 2 3 4 f(x) = e x and g(x) = lnx 356 If f(x) = x 2 − 6, find f −1 (x). 350 Which expression is equivalent to (n m p)(x), 1 3 2x 2 + 17x + 36 2x 2 + 17x + 66 2x 2 − 3x + 6 2x 2 − 3x + 36 353 If f(x) = 2x 2 + 1 and g(x) = 3x − 2, what is the value of f(g( − 2))? 1 −127 2 −23 3 25 4 129 354 If f(x) = x 2 − x and g(x) = x + 1, determine f(g(x)) in simplest form. 46 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org A2.A.46: TRANSFORMATIONS WITH FUNCTIONS AND RELATIONS 360 The minimum point on the graph of the equation y = f(x) is (−1,−3). What is the minimum point on the graph of the equation y = f(x) + 5 ? 1 (−1,2) 2 (−1,−8) 3 (4,−3) 4 (−6,−3) 359 The graph below shows the function f(x). 361 The function f(x) is graphed on the set of axes below. On the same set of axes, graph f(x + 1) + 2. Which graph represents the function f(x + 2)? 1 362 Which transformation of y = f(x) moves the graph 7 units to the left and 3 units down? 1 y = f(x + 7) − 3 2 y = f(x + 7) + 3 3 y = f(x − 7) − 3 4 y = f(x − 7) + 3 2 3 4 47 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 368 What is the common difference of the arithmetic sequence 5,8,11,14? 8 1 5 2 −3 3 3 4 9 SEQUENCES AND SERIES A2.A.29-33: SEQUENCES 363 What is the formula for the nth term of the sequence 54,18,6,. . .? n 1 1 a n = 6 3 2 1 n − 1 a n = 6 3 3 1 n a n = 54 3 4 369 Which arithmetic sequence has a common difference of 4? 1 {0,4n,8n,12n,. . . } 2 {n,4n,16n,64n,. . . } 3 {n + 1,n + 5,n + 9,n + 13,. . . } 4 {n + 4,n + 16,n + 64,n + 256,. . . } 1 n − 1 a n = 54 3 370 What is the common difference in the sequence 2a + 1, 4a + 4, 6a + 7, 8a + 10, . . .? 1 2a + 3 2 −2a − 3 3 2a + 5 4 −2a + 5 364 What is a formula for the nth term of sequence B shown below? B = 10,12,14,16,. . . 1 b n = 8 + 2n 2 b n = 10 + 2n 3 b n = 10(2) n 4 b n = 10(2) n − 1 371 What is the common difference of the arithmetic sequence below? −7x,−4x,−x,2x,5x,. . . 1 −3 2 −3x 3 3 4 3x 365 A sequence has the following terms: a 1 = 4, a 2 = 10, a 3 = 25, a 4 = 62.5. Which formula represents the nth term in the sequence? 1 a n = 4 + 2.5n 2 a n = 4 + 2.5(n − 1) 3 a n = 4(2.5) n 4 a n = 4(2.5) n − 1 372 What is the common ratio of the geometric sequence whose first term is 27 and fourth term is 64? 3 1 4 64 2 81 4 3 3 37 4 3 366 In an arithmetic sequence, a 4 = 19 and a 7 = 31. Determine a formula for a n , the n th term of this sequence. 367 A theater has 35 seats in the first row. Each row has four more seats than the row before it. Which expression represents the number of seats in the nth row? 1 35 + (n + 4) 2 35 + (4n) 3 35 + (n + 1)(4) 4 35 + (n − 1)(4) 48 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 373 What is the common ratio of the geometric sequence shown below? −2,4,−8,16,. . . 1 1 − 2 2 2 3 −2 4 −6 378 An arithmetic sequence has a first term of 10 and a sixth term of 40. What is the 20th term of this sequence? 1 105 2 110 3 124 4 130 379 Find the first four terms of the recursive sequence defined below. a 1 = −3 374 What is the common ratio of the sequence 1 5 3 3 3 4 9 a b ,− a b , ab 5 ,. . .? 64 32 16 3b 1 − 2 2a 6b 2 − 2 a 3 4 a n = a (n − 1) − n 380 Find the third term in the recursive sequence a k + 1 = 2a k − 1, where a 1 = 3. 3a 2 b 6a 2 − b − 381 Use the recursive sequence defined below to express the next three terms as fractions reduced to lowest terms. a1 = 2 −2 a n = 3 a n − 1 1 3 9 375 The common ratio of the sequence − , ,− is 2 4 8 3 1 − 2 2 2 − 3 1 3 − 2 1 4 − 4 382 What is the fourth term of the sequence defined by a 1 = 3xy 5 2x a n = a n − 1 ? y 1 12x 3 y 3 2 24x 2 y 4 3 24x 4 y 2 4 48x 5 y 376 What is the fifteenth term of the sequence 5,−10,20,−40,80,. . .? 1 −163,840 2 −81,920 3 81,920 4 327,680 383 The first four terms of the sequence defined by 1 a 1 = and a n + 1 = 1 − a n are 2 1 1 1 1 , , , 1 2 2 2 2 1 1 ,1,1 ,2 2 2 2 1 1 1 1 , , , 3 2 4 8 16 1 1 1 1 ,1 ,2 ,3 4 2 2 2 2 377 What is the fifteenth term of the geometric sequence − 5, 10 ,−2 5 ,. . .? 1 2 3 4 −128 5 128 10 −16384 5 16384 10 49 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org A2.N.10, A.34: SIGMA NOTATION 2 390 What is the value of ∑ (n + 2 n ) is 2 1 2 3 4 n =0 1 2 3 4 x x=0 2 384 The value of the expression 2 ∑ (3 − 2a) ? 12 22 24 26 4a − 2a + 12 4a 2 − 2a + 13 4a 2 − 14a + 12 4a 2 − 14a + 13 2 4 391 Simplify: 5 385 Evaluate: 10 + ∑ (n 3 ∑ x − a a=1 − 1) 2 . n=1 392 Mrs. Hill asked her students to express the sum 1 + 3 + 5 + 7 + 9 +. . .+ 39 using sigma notation. Four different student answers were given. Which student answer is correct? 5 386 The value of the expression ∑ (−r 2 + r) is r =3 1 2 3 4 −38 −12 26 62 20 1 k=1 40 2 ∑ (−n ∑ (k − 1) k=2 3 387 Evaluate: ∑ (2k − 1) 4 − n) 37 n=1 3 ∑ (k + 2) k = −1 39 5 388 The expression 4 + ∑ 3(k − x) is equal to 4 1 2 3 4 58 − 4x 46 − 4x 58 − 12x 46 − 12x 393 Express the sum 7 + 14 + 21 + 28 +. . .+ 105 using sigma notation. 394 Which summation represents 5 + 7 + 9 + 11 +. . .+ 43? 4 389 Which expression is equivalent to ∑ (a − n) ? 2 43 n=1 1 2 3 4 ∑ (2k − 1) k=1 k=2 1 2a + 17 4a 2 + 30 2a 2 − 10a + 17 4a 2 − 20a + 30 ∑n n=5 2 20 2 ∑ (2n + 3) n=1 24 3 ∑ (2n − 3) n=4 23 4 ∑ (3n − 4) n=3 50 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 398 What is the sum of the first 19 terms of the sequence 3,10,17,24,31,. . .? 1 1188 2 1197 3 1254 4 1292 1 2 mile on day 1, and mile on day 2, 3 3 1 2 and 1 miles on day 3, and 2 miles on day 4, and 3 3 this pattern continued for 3 more days. Which expression represents the total distance the jogger ran? 395 A jogger ran 399 Determine the sum of the first twenty terms of the sequence whose first five terms are 5, 14, 23, 32, 41. 7 1 ∑ 13 (2) d − 1 d=1 7 2 ∑ 13 (2) 400 The sum of the first eight terms of the series 3 − 12 + 48 − 192 + . . . is 1 −13,107 2 −21,845 3 −39,321 4 −65,535 d d=1 7 3 d−1 ∑ 2 13 d ∑ 2 13 d=1 d=1 7 4 TRIGONOMETRY 396 Which expression is equivalent to the sum of the sequence 6,12,20,30? A2.A.55: TRIGONOMETRIC RATIOS 401 In the diagram below of right triangle KTW, KW = 6, KT = 5, and m∠KTW = 90. 7 1 ∑2 n − 10 n=4 6 2 ∑ 2n3 2 n=3 5 3 ∑ 5n − 4 n=2 5 4 ∑n 2 +n n =2 What is the measure of ∠K , to the nearest minute? 1 33°33' 2 33°34' 3 33°55' 4 33°56' A2.A.35: SERIES 397 An auditorium has 21 rows of seats. The first row has 18 seats, and each succeeding row has two more seats than the previous row. How many seats are in the auditorium? 1 540 2 567 3 760 4 798 51 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 402 Which ratio represents csc A in the diagram below? 1 2 3 4 404 In the diagram below, the length of which line segment is equal to the exact value of sin θ ? 25 24 25 7 24 7 7 24 1 2 3 4 403 In the diagram below of right triangle JTM, JT = 12, JM = 6, and m∠JMT = 90. TO TS OR OS 405 In the right triangle shown below, what is the measure of angle S, to the nearest minute? 1 2 3 4 What is the value of cot J ? 3 3 1 2 3 4 2 28°1' 28°4' 61°56' 61°93' 406 By law, a wheelchair service ramp may be inclined no more than 4.76°. If the base of a ramp begins 15 feet from the base of a public building, which equation could be used to determine the maximum height, h, of the ramp where it reaches the building’s entrance? h 1 sin 4.76° = 15 15 2 sin 4.76° = h h 3 tan4.76° = 15 15 4 tan4.76° = h 3 2 3 3 52 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 413 What is the number of degrees in an angle whose measure is 2 radians? 360 1 A2.M.1-2: RADIAN MEASURE 407 What is the radian measure of the smaller angle formed by the hands of a clock at 7 o’clock? 1 2 3 4 π 2 2 2π 3 5π 6 7π 6 3 4 π π 360 360 90 414 Find, to the nearest tenth, the radian measure of 216º. 415 Convert 3 radians to degrees and express the answer to the nearest minute. 4π 408 The terminal side of an angle measuring 5 radians lies in Quadrant 1 I 2 II 3 III 4 IV 416 What is the number of degrees in an angle whose 8π radian measure is ? 5 1 576 2 288 3 225 4 113 409 Find, to the nearest minute, the angle whose measure is 3.45 radians. 417 Approximately how many degrees does five radians equal? 1 286 2 900 410 What is the number of degrees in an angle whose 11π ? radian measure is 12 1 150 2 165 3 330 4 518 3 4 π 36 5π 418 Convert 2.5 radians to degrees, and express the answer to the nearest minute. 411 What is the radian measure of an angle whose measure is −420°? 7π 1 − 3 7π 2 − 6 7π 3 6 7π 4 3 419 Determine, to the nearest minute, the degree 5 π radians. measure of an angle of 11 420 Determine, to the nearest minute, the number of degrees in an angle whose measure is 2.5 radians. 412 Find, to the nearest tenth of a degree, the angle whose measure is 2.5 radians. 53 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 422 In which graph is θ coterminal with an angle of −70°? A2.A.60: UNIT CIRCLE 421 On the unit circle shown in the diagram below, sketch an angle, in standard position, whose degree measure is 240 and find the exact value of sin240°. 1 2 3 4 54 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 423 If m∠θ = −50 , which diagram represents θ drawn in standard position? 425 If sin θ < 0 and cot θ > 0, in which quadrant does the terminal side of angle θ lie? 1 I 2 II 3 III 4 IV A2.A.56, 62, 66: DETERMINING TRIGONOMETRIC FUNCTIONS 1 426 In the interval 0° ≤ x < 360°, tanx is undefined when x equals 1 0º and 90º 2 90º and 180º 3 180º and 270º 4 90º and 270º 427 Express the product of cos 30° and sin 45° in simplest radical form. 2 428 If θ is an angle in standard position and its terminal side passes through the point (−3,2), find the exact value of csc θ . 429 Angle θ is in standard position and (−4,0) is a point on the terminal side of θ . What is the value of sec θ ? 1 −4 2 −1 3 0 4 undefined 3 430 Circle O has a radius of 2 units. An angle with a π radians is in standard position. If 6 the terminal side of the angle intersects the circle at point B, what are the coordinates of B? 3 1 , 1 2 2 2 3,1 3 1 3 , 2 2 4 1, 3 measure of 4 A2.A.60: FINDING THE TERMINAL SIDE OF AN ANGLE 424 An angle, P, drawn in standard position, terminates in Quadrant II if 1 cos P < 0 and csc P < 0 2 sinP > 0 and cos P > 0 3 csc P > 0 and cot P < 0 4 tanP < 0 and sec P > 0 55 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 431 The value of tan126°43′ to the nearest ten-thousandth is 1 −1.3407 2 −1.3408 3 −1.3548 4 −1.3549 435 In the diagram below of a unit circle, the ordered 2 2 represents the point where the ,− pair − 2 2 terminal side of θ intersects the unit circle. 432 Which expression, when rounded to three decimal places, is equal to −1.155? 5π 1 sec 6 2 3 4 tan(49°20 ′) 3π sin − 5 csc(−118°) 433 The value of csc 138°23′ rounded to four decimal places is 1 −1.3376 2 −1.3408 3 1.5012 4 1.5057 What is m∠θ ? 1 45 2 135 3 225 4 240 A2.A.64: USING INVERSE TRIGONOMETRIC FUNCTIONS 3 ? 434 What is the principal value of cos − 2 1 −30° 2 60° 3 150° 4 240° −1 5 436 If sin −1 = A, then 8 5 1 sin A = 8 8 2 sin A = 5 5 3 cos A = 8 8 4 cos A = 5 3 3 = 437 If tan Arc cos , then k is k 3 1 1 2 2 3 2 4 56 3 2 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org A2.A.61: ARC LENGTH 7 and ∠A terminates in Quadrant IV, 25 tan A equals 7 1 − 25 7 2 − 24 24 3 − 7 24 4 − 25 438 If sin A = − 442 A circle has a radius of 4 inches. In inches, what is the length of the arc intercepted by a central angle of 2 radians? 1 2π 2 2 3 8π 4 8 443 A circle is drawn to represent a pizza with a 12 inch diameter. The circle is cut into eight congruent pieces. What is the length of the outer edge of any one piece of this circle? 3π 1 4 2 π 3π 3 2 4 3π 15 439 What is the value of tan Arc cos ? 17 8 1 15 8 2 17 15 3 8 17 4 8 444 Circle O shown below has a radius of 12 centimeters. To the nearest tenth of a centimeter, determine the length of the arc, x, subtended by an angle of 83°50'. A2.A.57: REFERENCE ANGLES 440 Expressed as a function of a positive acute angle, cos(−305°) is equal to 1 −cos 55° 2 cos 55° 3 −sin 55° 4 sin 55° 441 Expressed as a function of a positive acute angle, sin230° is equal to 1 −sin 40° 2 −sin 50° 3 sin 40° 4 sin 50° 445 A wheel has a radius of 18 inches. Which distance, to the nearest inch, does the wheel travel when it 2π radians? rotates through an angle of 5 1 45 2 23 3 13 4 11 57 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 446 In a circle, an arc length of 6.6 is intercepted by a 2 central angle of radians. Determine the length of 3 the radius. 453 Which trigonometric expression does not simplify to 1? 1 sin 2 x(1 + cot 2 x) 2 3 A2.A.58-59: COFUNCTION AND RECIPROCAL TRIGONOMETRIC FUNCTIONS 447 If ∠A is acute and tanA = 1 2 3 4 2 , then 3 454 Show that 2 3 1 cot A = 3 cot A = sec 2 x − 1 is equivalent to sin 2 x. 2 sec x 455 Express the exact value of csc 60°, with a rational denominator. 2 3 1 cot(90° − A) = 3 456 The exact value of csc 120° is cot(90° − A) = 448 The expression 1 2 3 4 4 sec 2 x(1 − sin 2 x) cos 2 x(tan 2 x − 1) cot 2 x(sec 2 x − 1) 2 2 3 3 2 3 − 1 sin 2 θ + cos 2 θ is equivalent to 1 − sin 2 θ 4 cos 2 θ sin 2 θ sec 2 θ csc 2 θ 2 3 3 −2 A2.A.67: SIMPLIFYING TRIGONOMETRIC EXPRESSIONS & PROVING TRIGONOMETRIC IDENTITIES 449 Express cos θ(sec θ − cos θ), in terms of sin θ . 457 Which expression always equals 1? 1 cos 2 x − sin 2 x 2 cos 2 x + sin 2 x 3 cos x − sinx 4 cos x + sinx 450 If sec(a + 15)° =csc(2a)°, find the smallest positive value of a, in degrees. cot x sin x as a single trigonometric sec x function, in simplest form, for all values of x for which it is defined. 451 Express 458 Starting with sin 2 A + cos 2 A = 1, derive the formula tan 2 A + 1 = sec 2 A . 459 Show that sec θ sin θ cot θ = 1 is an identity. cot x is equivalent to 452 The expression csc x 1 sinx 2 cos x 3 tanx 4 sec x A2.A.76: ANGLE SUM AND DIFFERENCE IDENTITIES 460 The expression cos 4x cos 3x + sin4x sin 3x is equivalent to 1 sinx 2 sin 7x 3 cos x 4 cos 7x 58 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org A2.A.77: DOUBLE AND HALF ANGLE IDENTITIES 5 and angles A and B 41 are in Quadrant I, find the value of tan(A + B) . 461 If tanA = 2 and sin B = 3 467 The expression cos 2 θ − cos 2θ is equivalent to 1 sin 2 θ 2 −sin 2 θ 3 cos 2 θ + 1 4 −cos 2 θ − 1 462 Express as a single fraction the exact value of sin75°. 463 Given angle A in Quadrant I with sin A = 12 and 13 3 angle B in Quadrant II with cos B = − , what is the 5 value of cos(A − B)? 33 1 65 33 2 − 65 63 3 65 63 4 − 65 468 If sin A = of sin2A? 464 The value of sin(180 + x) is equivalent to 1 −sinx 2 −sin(90 − x) 3 sinx 4 sin(90 − x) 1 2 5 3 2 2 5 9 3 4 5 9 4 − 4 5 9 1 469 What is a positive value of tan x , when 2 sin x = 0.8? 1 0.5 2 0.4 3 0.33 4 0.25 465 The expression sin(θ + 90)° is equivalent to 1 −sin θ 2 −cos θ 3 sin θ 4 cos θ 1 470 If sin A = , what is the value of cos 2A? 3 2 1 − 3 2 2 3 7 3 − 9 7 4 9 466 If sin x = siny = a and cos x = cos y = b , then cos(x − y) is 1 2 3 4 2 where 0° < A < 90°, what is the value 3 b2 − a2 b2 + a2 2b − 2a 2b + 2a 59 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 477 What is the solution set for 2cos θ − 1 = 0 in the interval 0° ≤ θ < 360°? 1 {30°,150°} 2 {60°,120°} 3 {30°,330°} 4 {60°,300°} 3 471 If sin A = , what is the value of cos 2A? 8 9 1 − 64 1 2 4 23 3 32 55 4 64 472 The expression 1 2 3 4 2 3 4 1 + cos 2A is equivalent to sin2A cot A tanA sec A 1 + cot 2A 473 If cos θ = 1 478 What is the solution set of the equation − 2 sec x = 2 when 0° ≤ x < 360°? 1 {45°,135°,225°,315°} 2 {45°,315°} 3 {135°,225°} 4 {225°,315°} 479 Find, algebraically, the measure of the obtuse angle, to the nearest degree, that satisfies the equation 5csc θ = 8. 3 , then what is cos 2θ ? 4 480 Solve algebraically for all exact values of x in the interval 0 ≤ x < 2π : 2sin2 x + 5sin x = 3 1 8 9 16 1 − 8 3 2 481 Solve sec x − 2 = 0 algebraically for all values of x in 0° ≤ x < 360°. 482 In the interval 0° ≤ θ < 360°, solve the equation 5cos θ = 2sec θ − 3 algebraically for all values of θ , to the nearest tenth of a degree. 483 Which values of x in the interval 0° ≤ x < 360° satisfy the equation 2sin 2 x + sin x − 1 = 0? 1 {30°,270°} 2 {30°,150°,270°} 3 {90°,210°,330°} 4 {90°,210°,270°,330°} A2.A.68: TRIGONOMETRIC EQUATIONS 474 What are the values of θ in the interval 0° ≤ θ < 360° that satisfy the equation tan θ − 3 = 0? 1 60º, 240º 2 72º, 252º 3 72º, 108º, 252º, 288º 4 60º, 120º, 240º, 300º 475 Find all values of θ in the interval 0° ≤ θ < 360° that satisfy the equation sin2θ = sin θ . 476 Solve the equation 2tan C − 3 = 3tan C − 4 algebraically for all values of C in the interval 0° ≤ C < 360°. 60 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 489 What is the period of the graph of the equation 1 y = sin2x ? 3 1 1 3 2 2 3 π 4 6π A2.A.69: PROPERTIES OF TRIGONOMETRIC FUNCTIONS 484 What is the period of the function 1 x y = sin − π ? 2 3 1 2 3 4 1 2 1 3 2 π 3 6π A2.A.72: IDENTIFYING THE EQUATION OF A TRIGONOMETRIC GRAPH 490 Which equation is graphed in the diagram below? 485 What is the period of the function f(θ) = −2cos 3θ ? 1 π 2π 2 3 3π 3 2 4 2π 486 Which equation represents a graph that has a period of 4π ? 1 1 y = 3 sin x 2 2 y = 3 sin2x 1 3 y = 3 sin x 4 4 y = 3 sin4x 487 What is the period of the graph y = 1 2 3 4 π 1 2 3 1 sin6x ? 2 4 6 π 3 π 2 6π 488 How many full cycles of the function y = 3 sin2x appear in radians? 1 1 2 2 3 3 4 4 61 π y = 3 cos x + 8 30 π y = 3 cos x + 5 15 π y = −3 cos x + 8 30 π y = −3 cos x + 5 15 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 491 Write an equation for the graph of the trigonometric function shown below. 493 Which equation represents the graph below? 1 2 3 4 492 Which equation is represented by the graph below? y = −2sin 2x 1 y = −2 sin x 2 y = −2 cos 2x 1 y = −2 cos x 2 494 The periodic graph below can be represented by the trigonometric equation y = a cos bx + c where a, b, and c are real numbers. 1 2 3 4 y = 2 cos 3x y = 2 sin3x 2π y = 2 cos x 3 2π y = 2 sin x 3 State the values of a, b, and c, and write an equation for the graph. 62 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 496 Which graph shows y = cos −1 x ? A2.A.65, 70-71: GRAPHING TRIGONOMETRIC FUNCTIONS 495 Which graph represents the equation y = cos −1 x ? 1 1 2 2 3 3 4 4 63 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic 497 Which graph represents one complete cycle of the equation y = sin3π x ? 498 Which equation is represented by the graph below? 1 2 1 2 3 4 3 4 64 y = cot x y = csc x y = sec x y = tanx Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 500 Which is a graph of y = cot x ? 499 Which equation is sketched in the diagram below? 1 1 2 3 4 y = csc x y = sec x y = cot x y = tanx 2 3 4 A2.A.63: DOMAIN AND RANGE 501 The function f(x) = tanx is defined in such a way that f − 1 (x) is a function. What can be the domain of f(x)? 1 {x | 0 ≤ x ≤ π } 2 {x | 0 ≤ x ≤ 2π } π π 3 x | − < x < 2 2 3π π 4 x | − < x < 2 2 65 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 507 In parallelogram BFLO, OL = 3.8, LF = 7.4, and m∠O = 126. If diagonal BL is drawn, what is the area of BLF ? 1 11.4 2 14.1 3 22.7 4 28.1 502 In which interval of f(x) = cos(x) is the inverse also a function? 1 2 3 4 − − π 2 π <x< π 2 π ≤x≤ 2 2 0≤x≤π 3π π ≤x≤ 2 2 508 The two sides and included angle of a parallelogram are 18, 22, and 60°. Find its exact area in simplest form. 503 Which statement regarding the inverse function is true? 1 A domain of y = sin −1 x is [0,2π ]. 509 The area of triangle ABC is 42. If AB = 8 and m∠B = 61, the length of BC is approximately 1 5.1 2 9.2 3 12.0 4 21.7 −1 2 The range of y = sin x is [−1,1]. 3 A domain of y = cos −1 x is (−∞,∞). 4 The range of y = cos −1 x is [0, π ]. A2.A.74: USING TRIGONOMETRY TO FIND AREA 510 A ranch in the Australian Outback is shaped like triangle ACE, with m∠A = 42, m∠E = 103 , and AC = 15 miles. Find the area of the ranch, to the nearest square mile. 504 In ABC , m∠A = 120 , b = 10, and c = 18. What is the area of ABC to the nearest square inch? 1 52 2 78 3 90 4 156 511 Find, to the nearest tenth of a square foot, the area of a rhombus that has a side of 6 feet and an angle of 50°. 505 Two sides of a parallelogram are 24 feet and 30 feet. The measure of the angle between these sides is 57°. Find the area of the parallelogram, to the nearest square foot. 512 Two sides of a triangular-shaped sandbox measure 22 feet and 13 feet. If the angle between these two sides measures 55°, what is the area of the sandbox, to the nearest square foot? 1 82 2 117 3 143 4 234 506 The sides of a parallelogram measure 10 cm and 18 cm. One angle of the parallelogram measures 46 degrees. What is the area of the parallelogram, to the nearest square centimeter? 1 65 2 125 3 129 4 162 513 The area of a parallelogram is 594, and the lengths of its sides are 32 and 46. Determine, to the nearest tenth of a degree, the measure of the acute angle of the parallelogram. 66 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 514 What is the area of a parallelogram that has sides measuring 8 cm and 12 cm and includes an angle of 120°? 1 24 3 2 3 4 518 In 1 2 48 3 83 3 96 3 3 4 A2.A.73: LAW OF SINES 515 In ABC , m∠A = 32, a = 12, and b = 10. Find the measures of the missing angles and side of ABC . Round each measure to the nearest tenth. PQR, p equals r sinP sinQ r sinP sinR r sinR sinP q sinR sinQ A2.A.75: LAW OF SINES-THE AMBIGUOUS CASE 519 In ABC , m∠A = 74, a = 59.2, and c = 60.3. What are the two possible values for m∠C , to the nearest tenth? 1 73.7 and 106.3 2 73.7 and 163.7 3 78.3 and 101.7 4 78.3 and 168.3 516 The diagram below shows the plans for a cell phone tower. A guy wire attached to the top of the tower makes an angle of 65 degrees with the ground. From a point on the ground 100 feet from the end of the guy wire, the angle of elevation to the top of the tower is 32 degrees. Find the height of the tower, to the nearest foot. 520 How many distinct triangles can be formed if m∠A = 35, a = 10, and b = 13? 1 1 2 2 3 3 4 0 521 Given ABC with a = 9, b = 10, and m∠B = 70, what type of triangle can be drawn? 1 an acute triangle, only 2 an obtuse triangle, only 3 both an acute triangle and an obtuse triangle 4 neither an acute triangle nor an obtuse triangle 517 As shown in the diagram below, fire-tracking station A is 100 miles due west of fire-tracking station B. A forest fire is spotted at F, on a bearing 47° northeast of station A and 15° northeast of station B. Determine, to the nearest tenth of a mile, the distance the fire is from both station A and station B. [N represents due north.] 522 In MNP , m = 6 and n = 10. Two distinct triangles can be constructed if the measure of angle M is 1 35 2 40 3 45 4 50 67 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 523 In KLM , KL = 20 , LM = 13, and m∠K = 40 . The measure of ∠M ? 1 must be between 0° and 90° 2 must equal 90° 3 must be between 90° and 180° 4 is ambiguous 529 In ABC , a = 15, b = 14, and c = 13, as shown in the diagram below. What is the m∠C , to the nearest degree? 524 In DEF , d = 5, e = 8, and m∠D = 32. How many distinct triangles can be drawn given these measurements? 1 1 2 2 3 3 4 0 525 How many distinct triangles can be constructed if m∠A = 30, side a = 34 , and side b = 12? 1 one acute triangle 2 one obtuse triangle 3 two triangles 4 none 1 2 3 4 526 In triangle ABC, determine the number of distinct triangles that can be formed if m∠A = 85, side a = 8, and side c = 2. Justify your answer. 53 59 67 127 530 Two sides of a parallelogram measure 27 cm and 32 cm. The included angle measures 48°. Find the length of the longer diagonal of the parallelogram, to the nearest centimeter. A2.A.73: LAW OF COSINES 531 In FGH , f = 6, g = 9 , and m∠H = 57. Which statement can be used to determine the numerical value of h? 1 h 2 = 6 2 + 9 2 − 2(9)(h) cos 57° 527 In a triangle, two sides that measure 6 cm and 10 cm form an angle that measures 80°. Find, to the nearest degree, the measure of the smallest angle in the triangle. 2 528 In 1 2 3 4 ABC , a = 3, b = 5, and c = 7. What is m∠C ? 22 38 60 120 3 4 h 2 = 6 2 + 9 2 − 2(6)(9) cos 57° 6 2 = 9 2 + h 2 − 2(9)(h) cos 57° 9 2 = 6 2 + h 2 − 2(6)(h) cos 57° 532 Find the measure of the smallest angle, to the nearest degree, of a triangle whose sides measure 28, 47, and 34. 533 In a triangle, two sides that measure 8 centimeters and 11 centimeters form an angle that measures 82°. To the nearest tenth of a degree, determine the measure of the smallest angle in the triangle. 68 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org A2.A.73: VECTORS 540 Which equation represents a circle with its center at (2,−3) and that passes through the point (6,2)? 534 Two forces of 25 newtons and 85 newtons acting on a body form an angle of 55°. Find the magnitude of the resultant force, to the nearest hundredth of a newton. Find the measure, to the nearest degree, of the angle formed between the resultant and the larger force. 1 (x − 2) 2 + (y + 3) 2 = 2 (x + 2) + (y − 3) = 41 (x − 2) 2 + (y + 3) 2 = 41 (x + 2) 2 + (y − 3) 2 = 41 3 4 2 41 2 541 Write an equation of the circle shown in the graph below. 535 The measures of the angles between the resultant and two applied forces are 60° and 45°, and the magnitude of the resultant is 27 pounds. Find, to the nearest pound, the magnitude of each applied force. 536 Two forces of 40 pounds and 28 pounds act on an object. The angle between the two forces is 65°. Find the magnitude of the resultant force, to the nearest pound. Using this answer, find the measure of the angle formed between the resultant and the smaller force, to the nearest degree. CONICS A2.A.47-49: EQUATIONS OF CIRCLES 537 The equation x 2 + y 2 − 2x + 6y + 3 = 0 is equivalent to 1 (x − 1) 2 + (y + 3) 2 = −3 2 3 4 542 A circle shown in the diagram below has a center of (−5,3) and passes through point (−1,7). (x − 1) 2 + (y + 3) 2 = 7 (x + 1) 2 + (y + 3) 2 = 7 (x + 1) 2 + (y + 3) 2 = 10 538 What are the coordinates of the center of a circle whose equation is x 2 + y 2 − 16x + 6y + 53 = 0 ? 1 (−8,−3) 2 (−8,3) 3 (8,−3) 4 (8,3) 539 What is the equation of the circle passing through the point (6,5) and centered at (3,−4)? 1 2 3 4 (x − 6) 2 + (y − 5) 2 (x − 6) 2 + (y − 5) 2 (x − 3) 2 + (y + 4) 2 (x − 3) 2 + (y + 4) 2 = 82 = 90 = 82 = 90 Write an equation that represents the circle. 69 Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic www.jmap.org 543 Which equation represents the circle shown in the graph below that passes through the point (0,−1)? 545 Which equation is represented by the graph below? 1 2 1 2 3 4 (x − 3) + (y + 4) (x − 3) 2 + (y + 4) 2 (x + 3) 2 + (y − 4) 2 (x + 3) 2 + (y − 4) 2 2 2 = 16 = 18 = 16 = 18 3 4 (x − 3) 2 + (y + 1) 2 (x + 3) 2 + (y − 1) 2 (x − 1) 2 + (y + 3) 2 (x + 3) 2 + (y − 1) 2 =5 =5 = 13 = 13 546 A circle with center O and passing through the origin is graphed below. 544 Write an equation of the circle shown in the diagram below. What is the equation of circle O? 1 x2 + y2 = 2 5 2 x 2 + y 2 = 20 3 (x + 4) 2 + (y − 2) 2 = 2 5 (x + 4) 2 + (y − 2) 2 = 20 4 70 ID: A Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic Answer Section 1 ANS: Controlled experiment because Howard is comparing the results obtained from an experimental sample against a control sample. PTS: 2 REF: 081030a2 STA: A2.S.1 TOP: Analysis of Data 2 ANS: 4 PTS: 2 REF: 011127a2 STA: A2.S.1 TOP: Analysis of Data 3 ANS: 4 PTS: 2 REF: 061101a2 STA: A2.S.1 TOP: Analysis of Data 4 ANS: 2 PTS: 2 REF: 061301a2 STA: A2.S.1 TOP: Analysis of Data 5 ANS: 4 PTS: 2 REF: 011406a2 STA: A2.S.1 TOP: Analysis of Data 6 ANS: 4 Students entering the library are more likely to spend more time studying, creating bias. PTS: 2 REF: fall0904a2 7 ANS: 4 PTS: 2 TOP: Analysis of Data 8 ANS: 1 PTS: 2 TOP: Analysis of Data 9 ANS: 4 PTS: 2 TOP: Average Known with Missing Data 10 ANS: 4 4 ⋅ 0 + 6 ⋅ 1 + 10 ⋅ 2 + 0 ⋅ 3 + 4k + 2 ⋅ 5 =2 4 + 6 + 10 + 0 + k + 2 STA: A2.S.2 REF: 011201a2 TOP: Analysis of Data STA: A2.S.2 REF: 061401a2 STA: A2.S.2 REF: 061124a2 STA: A2.S.3 4k + 36 =2 k + 22 4k + 36 = 2k + 44 2k = 8 k =4 PTS: 2 REF: 061221a2 STA: A2.S.3 1 TOP: Average Known with Missing Data ID: A 11 ANS: 3 PTS: 2 REF: fall0924a2 STA: A2.S.4 KEY: range, quartiles, interquartile range, variance 12 ANS: 7.4 TOP: Dispersion PTS: 2 REF: 061029a2 STA: A2.S.4 KEY: basic, group frequency distributions 13 ANS: σ x = 14.9. x = 40. There are 8 scores between 25.1 and 54.9. TOP: Dispersion PTS: 4 REF: 061237a2 STA: A2.S.4 TOP: Dispersion KEY: advanced 14 ANS: Ordered, the heights are 71, 71, 72, 74, 74, 75, 78, 79, 79, 83. Q 1 = 72 and Q 3 = 79. 79 − 72 = 7. PTS: 2 REF: 011331a2 STA: A2.S.4 TOP: Dispersion KEY: range, quartiles, interquartile range, variance 15 ANS: σ x ≈ 6.2. 6 scores are within a population standard deviation of the mean. Q 3 − Q 1 = 41 − 37 = 4 x ≈ 38.2 PTS: 4 REF: 061338a2 KEY: advanced 16 ANS: Q 1 = 3.5 and Q 3 = 10.5. 10.5 − 3.5 = 7. STA: A2.S.4 TOP: Dispersion PTS: 2 REF: 011430a2 STA: A2.S.4 KEY: range, quartiles, interquartile range, variance 17 ANS: 2 12 − 7 = 5 TOP: Dispersion PTS: 2 REF: 011525a2 STA: A2.S.4 KEY: range, quartiles, interquartile range, variance TOP: Dispersion 2 ID: A 18 ANS: 5.17 84.46 ± 5.17 79.29 − 89.63 5 + 7 + 5 = 17 PTS: 4 REF: 061538a2 STA: A2.S.4 TOP: Dispersion KEY: advanced, group frequency distributions 19 ANS: 2 PTS: 2 REF: 081509a2 STA: A2.S.4 TOP: Dispersion KEY: basic, group frequency distributions 20 ANS: 3 PTS: 2 REF: 061127a2 STA: A2.S.6 TOP: Regression 21 ANS: y = 2.001x 2.298 , 1,009. y = 2.001(15) 2.298 ≈ 1009 PTS: 4 22 ANS: y = 10.596(1.586) x REF: fall0938a2 STA: A2.S.7 TOP: Power Regression PTS: 2 REF: 081031a2 STA: A2.S.7 23 ANS: y = 27.2025(1.1509) x . y = 27.2025(1.1509) 18 ≈ 341 TOP: Exponential Regression PTS: 4 REF: 011238a2 24 ANS: y = 180.377(0.954) x STA: A2.S.7 TOP: Exponential Regression PTS: 2 REF: 061231a2 STA: A2.S.7 25 ANS: y = 215.983(1.652) x . 215.983(1.652) 7 ≈ 7250 TOP: Exponential Regression PTS: 4 26 ANS: y = 0.488(1.116) x STA: A2.S.7 TOP: Exponential Regression PTS: 2 REF: 061429a2 STA: A2.S.7 27 ANS: y = 733.646(0.786) x 733.646(0.786) 12 ≈ 41 TOP: Exponential Regression PTS: 4 REF: 011536a2 28 ANS: 2 PTS: 2 TOP: Correlation Coefficient TOP: Exponential Regression STA: A2.S.8 REF: 011337a2 STA: A2.S.7 REF: 061021a2 3 ID: A 29 ANS: 1 (4) shows the strongest linear relationship, but if r < 0, b < 0. The Regents announced that a correct solution was not provided for this question and all students should be awarded credit. PTS: 2 30 ANS: 1 REF: 011223a2 STA: A2.S.8 TOP: Correlation Coefficient PTS: 2 REF: 061225a2 STA: A2.S.8 31 ANS: 2 Since the coefficient of t is greater than 0, r > 0. TOP: Correlation Coefficient . PTS: 2 REF: 011303a2 STA: A2.S.8 TOP: Correlation Coefficient 32 ANS: 1 PTS: 2 REF: 061316a2 STA: A2.S.8 TOP: Correlation Coefficient 33 ANS: r A ≈ 0.976 r B ≈ 0.994 Set B has the stronger linear relationship since r is higher. PTS: 34 ANS: TOP: 35 ANS: 2 REF: 061535a2 2 PTS: 2 Correlation Coefficient 1 PTS: 2 KEY: interval 36 ANS: 3 68% × 50 = 34 REF: fall0915a2 STA: A2.S.8 REF: 081502a2 TOP: Correlation Coefficient STA: A2.S.8 STA: A2.S.5 TOP: Normal Distributions PTS: 2 REF: 081013a2 STA: A2.S.5 TOP: Normal Distributions KEY: predict 37 ANS: 68% of the students are within one standard deviation of the mean. 16% of the students are more than one standard deviation above the mean. PTS: 2 KEY: percent REF: 011134a2 STA: A2.S.5 4 TOP: Normal Distributions ID: A 38 ANS: no. over 20 is more than 1 standard deviation above the mean. 0.159 ⋅ 82 ≈ 13.038 PTS: 2 REF: 061129a2 KEY: predict 39 ANS: 3 34.1% + 19.1% = 53.2% STA: A2.S.5 TOP: Normal Distributions PTS: 2 KEY: probability 40 ANS: 2 x ±σ STA: A2.S.5 TOP: Normal Distributions PTS: 2 REF: 011307a2 STA: A2.S.5 KEY: interval 41 ANS: 2 Top 6.7% = 1.5 s.d. + σ = 1.5(104) + 576 = 732 TOP: Normal Distributions REF: 011212a2 153 ± 22 131 − 175 PTS: 2 REF: 011420a2 STA: A2.S.5 TOP: Normal Distributions KEY: predict 42 ANS: Less than 60 inches is below 1.5 standard deviations from the mean. 0.067 ⋅ 450 ≈ 30 PTS: 2 REF: 061428a2 KEY: predict 43 ANS: 81 − 57 sd = =8 3 STA: A2.S.5 TOP: Normal Distributions PTS: 2 REF: 011534a2 KEY: mean and standard deviation 44 ANS: 4 91 − 82 = 2.5 sd 3.6 STA: A2.S.5 TOP: Normal Distributions PTS: KEY: 45 ANS: TOP: STA: A2.S.5 TOP: Normal Distributions REF: fall0925a2 STA: A2.S.10 57 + 8 = 65 81 − 2(8) = 65 2 REF: 081521a2 interval 4 PTS: 2 Permutations 5 ID: A 46 ANS: No. TENNESSEE: PTS: 4 47 ANS: P9 362,880 = = 3,780. VERMONT: 7 P 7 = 5,040 96 4!⋅ 2!⋅ 2! 9 REF: 061038a2 39,916,800. STA: A2.S.10 TOP: Permutations P 12 479,001,600 = = 39,916,800 12 3!⋅ 2! 12 PTS: 2 REF: 081035a2 STA: A2.S.10 TOP: Permutations 48 ANS: 1 8 × 8 × 7 × 1 = 448. The first digit cannot be 0 or 5. The second digit cannot be 5 or the same as the first digit. The third digit cannot be 5 or the same as the first or second digit. PTS: 2 49 ANS: 1 720 6 P6 = = 60 12 3!2! REF: 011125a2 STA: A2.S.10 TOP: Permutations PTS: 2 50 ANS: 10 P 10 REF: 011324a2 STA: A2.S.10 TOP: Permutations PTS: 2 REF: 061330a2 51 ANS: 4 PTS: 2 TOP: Permutations 52 ANS: 3 2!⋅ 2!⋅ 2! = 8 STA: A2.S.10 REF: 011409a2 TOP: Permutations STA: A2.S.10 PTS: 2 53 ANS: 1 11 P 11 STA: A2.S.10 TOP: Permutations STA: A2.S.10 TOP: Permutations STA: A2.S.10 TOP: Permutations STA: A2.S.10 TOP: Permutations 3!⋅ 3!⋅ 2! 2!2!2!2! PTS: 2 54 ANS: 1 9 P9 4!⋅ 2!⋅ 2! PTS: 2 55 ANS: 1 11 P 11 3!2!2!2! PTS: 2 = 3,628,800 = 50,400 72 REF: 061425a2 = 39,916,800 = 2,494,800 16 REF: 011518a2 = 362,880 = 3,780 96 REF: 061511a2 = 39,916,800 = 831,600 48 REF: 081512a2 6 ID: A 56 ANS: 2 15 C 8 = 6,435 PTS: 2 57 ANS: 1 10 C 4 = 210 REF: 081012a2 STA: A2.S.11 TOP: Combinations PTS: 2 58 ANS: 25 C 20 = 53,130 REF: 061113a2 STA: A2.S.11 TOP: Combinations PTS: 2 59 ANS: 4 15 C 5 = 3,003. REF: 011232a2 STA: A2.S.11 TOP: Combinations 25 C 5 = 25 C 20 = 53,130. 25 C 15 = 3,268,760. PTS: 2 60 ANS: 3 20 C 4 = 4,845 REF: 061227a2 STA: A2.S.11 TOP: Combinations PTS: 2 61 ANS: 3 9 C 3 = 84 REF: 011509a2 STA: A2.S.11 TOP: Combinations PTS: 62 ANS: TOP: 63 ANS: TOP: 64 ANS: TOP: 65 ANS: TOP: 66 ANS: TOP: 67 ANS: TOP: 68 ANS: TOP: 2 REF: 081513a2 STA: A2.S.11 3 PTS: 2 REF: 061007a2 Differentiating Permutations and Combinations 1 PTS: 2 REF: 011117a2 Differentiating Permutations and Combinations 1 PTS: 2 REF: 011310a2 Differentiating Permutations and Combinations 1 PTS: 2 REF: 061317a2 Differentiating Permutations and Combinations 2 PTS: 2 REF: 011417a2 Differentiating Permutations and Combinations 3 PTS: 2 REF: 061523a2 Differentiating Permutations and Combinations 4 PTS: 2 REF: 081526a2 Differentiating Permutations and Combinations 7 TOP: Combinations STA: A2.S.9 STA: A2.S.9 STA: A2.S.9 STA: A2.S.9 STA: A2.S.9 STA: A2.S.9 STA: A2.S.9 ID: A 69 ANS: 41,040. PTS: 2 REF: fall0935a2 70 ANS: 3 3 C 1 ⋅5 C 2 = 3 ⋅ 10 = 30 STA: A2.S.12 TOP: Sample Space PTS: 2 71 ANS: 2 STA: A2.S.12 TOP: Combinations REF: 061422a2 π 3 + 2π π 3 2π 3 1 = = 3 2π PTS: 2 REF: 011108a2 STA: A2.S.13 TOP: Geometric Probability 72 ANS: 0.167. 10 C 8 ⋅ 0.6 8 ⋅ 0.4 2 + 10 C 9 ⋅ 0.6 9 ⋅ 0.4 1 + 10 C 10 ⋅ 0.6 10 ⋅ 0.4 0 ≈ 0.167 PTS: 4 REF: 061036a2 STA: A2.S.15 TOP: Binomial Probability KEY: at least or at most 73 ANS: 26.2%. 10 C 8 ⋅ 0.65 8 ⋅ 0.35 2 + 10 C 9 ⋅ 0.65 9 ⋅ 0.35 1 + 10 C 10 ⋅ 0.65 10 ⋅ 0.35 0 ≈ 0.262 PTS: 4 REF: 081038a2 STA: A2.S.15 TOP: Binomial Probability KEY: at least or at most 74 ANS: 2 6 1 2 2 7 1 1 2 8 1 0 0.468. 8 C 6 ≈ 0.27313. 8 C 7 ≈ 0.15607. 8 C 8 ≈ 0.03902. 3 3 3 3 3 3 PTS: 4 REF: 011138a2 KEY: at least or at most STA: A2.S.15 8 TOP: Binomial Probability ID: A 75 ANS: 3 2 1 2 40 51 . 5 C 3 = 243 3 3 243 4 1 1 2 10 C 5 4 3 3 = 243 1 5 2 0 = 1 C 5 3 3 3 243 PTS: 4 REF: 061138a2 KEY: at least or at most 76 ANS: 4 5 2 3 1 225 = 3 C2 8 8 512 STA: A2.S.15 TOP: Binomial Probability PTS: 2 REF: 011221a2 KEY: spinner 77 ANS: 1 PTS: 2 TOP: Binomial Probability 78 ANS: 1 3 3 4 = 35 1 81 = 2835 7 C3 4 4 64 256 16384 STA: A2.S.15 TOP: Binomial Probability REF: 061223a2 KEY: modeling STA: A2.S.15 ≈ 0.173 PTS: 2 REF: 061335a2 STA: A2.S.15 KEY: exactly 79 ANS: 4 1 5 0 5 C 4 ⋅ 0.28 ⋅ 0.72 + 5 C 5 ⋅ 0.28 ⋅ 0.72 ≈ 0.024 TOP: Binomial Probability PTS: 4 REF: 011437a2 STA: A2.S.15 TOP: Binomial Probability KEY: at least or at most 80 ANS: 0 5 1 4 2 3 5 C 0 ⋅ 0.57 ⋅ 0.43 + 5 C 1 ⋅ 0.57 ⋅ 0.43 + 5 C 2 ⋅ 0.57 ⋅ 0.43 ≈ 0.37 PTS: 4 REF: 061438a2 KEY: at least or at most 81 ANS: 2 5 3 = 6 32 3 = 576 C 6 5 5 5 3125 5 15,625 PTS: 2 KEY: exactly REF: 011532a2 STA: A2.S.15 TOP: Binomial Probability STA: A2.S.15 TOP: Binomial Probability 9 ID: A 82 ANS: 1 1 3 2 = 3 ⋅ 1 ⋅ 9 = 27 C 3 1 4 4 4 16 64 PTS: 2 REF: 061530a2 KEY: exactly 83 ANS: 2 4 1 3 = 35 16 1 = 560 C 81 27 2187 7 4 3 3 STA: A2.S.15 TOP: Binomial Probability PTS: 2 KEY: exactly 84 ANS: 1 STA: A2.S.15 TOP: Binomial Probability REF: 081531a2 | 1 | 1 4a + 6 = 4a − 10. 4a + 6 = −4a + 10. | 4 + 6 | − 4 = −10 || 2 || 2 6 ≠ −10 8a = 4 8 − 2 ≠ −10 4 1 a= = 8 2 PTS: 2 REF: 011106a2 STA: A2.A.1 85 ANS: 2 x − 2 = 3x + 10 −6 is extraneous. x − 2 = −3x − 10 −12 = 2x 4x = −8 −6 = x x = −2 PTS: 2 REF: 061513a2 86 ANS: 1 6x − 7 ≤ 5 6x − 7 ≥ −5 6x ≤ 12 TOP: Absolute Value Equations STA: A2.A.1 TOP: Absolute Value Equations STA: A2.A.1 TOP: Absolute Value Inequalities STA: A2.A.1 TOP: Absolute Value Inequalities 6x ≥ 2 x≤2 x≥ 1 3 PTS: 2 KEY: graph 87 ANS: −3 |6 − x| < −15 REF: fall0905a2 . |6 − x | > 5 6 − x > 5 or 6 − x < −5 1 > x or 11 < x PTS: 2 KEY: graph REF: 061137a2 10 ID: A 88 ANS: 3 4x − 5 4x − 5 < −1 > 1 or 3 3 4x − 5 > 3 4x > 8 x>2 4x − 5 < −3 4x < 2 x< 1 2 PTS: 2 REF: 061209a2 KEY: graph 89 ANS: 3 − 2x ≥ 7 or 3 − 2x ≤ −7 −2x ≥ 4 x ≤ −2 2x > −4 x>3 x < −2 PTS: 2 REF: 061307a2 KEY: graph 91 ANS: −4x + 5 < 13 −4x + 5 > −13 −2 < x < 4.5 TOP: Absolute Value Inequalities STA: A2.A.1 TOP: Absolute Value Inequalities STA: A2.A.1 TOP: Absolute Value Inequalities STA: A2.A.1 TOP: Absolute Value Inequalities −4x > −18 x < 4.5 PTS: 2 REF: 011432a2 92 ANS: 2x − 3 > 5 or 2x − 3 < −5 2x > 8 2x < −2 x>4 x < −1 PTS: 2 STA: A2.A.1 x≥5 2x > 6 x > −2 TOP: Absolute Value Inequalities −2x ≤ −10 PTS: 2 REF: 011334a2 KEY: graph 90 ANS: 1 2x − 1 > 5. 2x − 1 < −5 −4x < 8 STA: A2.A.1 REF: 061430a2 11 ID: A 93 ANS: |3x − 5| < x + 17 3x − 5 < x + 17 and 3x − 5 > −x − 17 −3 < x < 11 2x < 22 4x > −12 x < 11 x > −3 PTS: 4 REF: 081538a2 94 ANS: −b c 11 3 = − . Product = − Sum a a 5 5 STA: A2.A.1 TOP: Absolute Value Inequalities PTS: 2 REF: 061030a2 STA: A2.A.20 95 ANS: 2 −b 4 2 c −12 = = . product: = = −2 sum: 6 3 6 a a TOP: Roots of Quadratics PTS: 2 96 ANS: TOP: Roots of Quadratics REF: 011209a2 3x 2 − 11x + 6 = 0. Sum STA: A2.A.20 −b 11 c 6 = . Product = = 2 3 a a 3 PTS: 2 REF: 011329a2 97 ANS: −b c 1 1 = − . Product = − Sum a a 12 2 STA: A2.A.20 TOP: Roots of Quadratics PTS: 2 98 ANS: 4 2x 2 − 7x − 5 = 0 REF: 061328a2 STA: A2.A.20 TOP: Roots of Quadratics REF: 061414a2 STA: A2.A.20 TOP: Roots of Quadratics PTS: 2 REF: 011517a2 100 ANS: −b −2 c k = . Product = Sum a a 3 3 STA: A2.A.20 TOP: Roots of Quadratics STA: A2.A.20 TOP: Roots of Quadratics c −5 = a 2 PTS: 2 99 ANS: 3 c −3 = a 4 PTS: 2 REF: 061534a2 12 ID: A 101 ANS: 2 c −12 P= = = −4 a 3 PTS: 2 REF: 081506a2 102 ANS: 3 −b −(−3) 3 c −8 = −2 S= = = . P= = 4 a 4 a 4 STA: A2.A.20 TOP: Roots of Quadratics PTS: 2 REF: fall0912a2 KEY: basic 103 ANS: 3 −b −6 c 4 = = −3. = =2 2 a a 2 STA: A2.A.21 TOP: Roots of Quadratics PTS: 2 KEY: basic 104 ANS: STA: A2.A.21 TOP: Roots of Quadratics x 2 − 6x − 27 = 0, REF: 011121a2 −b c = 6. = −27. If a = 1 then b = −6 and c = −27 a a PTS: 4 KEY: basic 105 ANS: 3 sum of the roots, REF: 061130a2 STA: A2.A.21 TOP: Roots of Quadratics −b −(−9) 9 c 3 = = . product of the roots, = 4 a 4 a 4 PTS: 2 REF: 061208a2 STA: A2.A.21 TOP: Roots of Quadratics KEY: basic 106 ANS: 3 −b −(−4) = = 4. If the sum is 4, the roots must be 7 and −3. a 1 PTS: 2 REF: 011418a2 STA: A2.A.21 KEY: advanced 107 ANS: 4 6x − x 3 − x 2 = −x(x 2 + x − 6) = −x(x + 3)(x − 2) TOP: Roots of Quadratics PTS: 2 REF: fall0917a2 STA: A2.A.7 KEY: single variable 108 ANS: 4 12x 4 + 10x 3 − 12x 2 = 2x 2 (6x 2 + 5x − 6) = 2x 2 (2x + 3)(3x − 2) TOP: Factoring Polynomials PTS: 2 REF: 061008a2 KEY: single variable STA: A2.A.7 13 TOP: Factoring Polynomials ID: A 109 ANS: 10ax 2 − 23ax − 5a = a(10x 2 − 23x − 5) = a(5x + 1)(2x − 5) PTS: 2 REF: 081028a2 STA: A2.A.7 KEY: multiple variables 110 ANS: 12t 8 − 75t 4 = 3t 4 (4t 4 − 25) = 3t 4 (2t 2 + 5)(2t 2 − 5) PTS: 2 REF: 061133a2 STA: A2.A.7 TOP: Factoring the Difference of Perfect Squares 111 ANS: 2 x 3 + 3x 2 − 4x − 12 TOP: Factoring Polynomials KEY: binomial x 2 (x + 3) − 4(x + 3) (x 2 − 4)(x + 3) (x + 2)(x − 2)(x + 3) PTS: 2 REF: 061214a2 112 ANS: 3 3x 3 − 5x 2 − 48x + 80 STA: A2.A.7 TOP: Factoring by Grouping REF: 011317a2 STA: A2.A.7 TOP: Factoring by Grouping REF: 011426a2 STA: A2.A.7 TOP: Factoring by Grouping REF: 011511a2 STA: A2.A.7 TOP: Factoring by Grouping x 2 (3x − 5) − 16(3x − 5) (x 2 − 16)(3x − 5) (x + 4)(x − 4)(3x − 5) PTS: 2 113 ANS: 4 x 2 (x + 2) − (x + 2) (x 2 − 1)(x + 2) (x + 1)(x − 1)(x + 2) PTS: 2 114 ANS: 2 x 3 − 2x 2 − 9x + 18 x 2 (x − 2) − 9(x − 2) (x 2 − 9)(x − 2) (x + 3)(x − 3)(x − 2) PTS: 2 14 ID: A 115 ANS: x 2 (x − 6) − 25(x − 6) (x 2 − 25)(x − 6) (x + 5)(x − 5)(x − 6) PTS: 2 116 ANS: 4 3± REF: 061532a2 REF: 061009a2 STA: A2.A.25 TOP: Quadratics with Irrational Solutions STA: A2.A.25 TOP: Quadratics with Irrational Solutions 7 2 − 4(2)(−3) −7 ± 73 = 2(2) 4 PTS: 2 118 ANS: 2± TOP: Factoring by Grouping (−3) 2 − 4(1)(−9) 3 ± 45 3 ± 3 5 = = 2(1) 2 2 PTS: 2 117 ANS: 3 −7 ± STA: A2.A.7 REF: 081009a2 (−2) 2 − 4(6)(−3) 2 ± 76 2 ± 4 19 2 ± 2 19 1 ± 19 = = = = 2(6) 12 12 12 6 PTS: 2 119 ANS: 2 REF: 011332a2 60 = −16t + 5t + 105 t = 2 −5 ± STA: A2.A.25 TOP: Quadratics with Irrational Solutions 5 2 − 4(−16)(45) −5 ± 53.89 ≈ ≈ 1.84 2(−16) −32 0 = −16t 2 + 5t + 45 PTS: 2 120 ANS: REF: 061424a2 (x + 14)(x + 22) = 800 x = −36 ± STA: A2.A.25 TOP: Quadratics with Irrational Solutions (−36) 2 − 4(1)(−492) −36 + 3264 = ≈ 10.6 10 feet increase. 2(1) 2 x 2 + 36x + 308 = 800 x 2 + 36x − 492 = 0 PTS: 6 REF: 011539a2 STA: A2.A.25 15 TOP: Quadratics with Irrational Solutions ID: A 121 ANS: b 2 − 4ac = 0 k 2 − 4(1)(4) = 0 k 2 − 16 = 0 (k + 4)(k − 4) = 0 k = ±4 PTS: 2 REF: 061028a2 STA: A2.A.2 KEY: determine equation given nature of roots 122 ANS: 4 b 2 − 4ac = 3 2 − 4(9)(−4) = 9 + 144 = 153 TOP: Using the Discriminant PTS: 2 REF: 081016a2 STA: A2.A.2 KEY: determine nature of roots given equation 123 ANS: 3 b 2 − 4ac = (−10) 2 − 4(1)(25) = 100 − 100 = 0 TOP: Using the Discriminant PTS: 2 REF: 011102a2 STA: A2.A.2 TOP: Using the Discriminant KEY: determine nature of roots given equation 124 ANS: 4 PTS: 2 REF: 011323a2 STA: A2.A.2 TOP: Using the Discriminant KEY: determine nature of roots given equation 125 ANS: 2 b 2 − 4ac = (−9) 2 − 4(2)(4) = 81 − 32 = 49 PTS: 2 REF: 011411a2 STA: A2.A.2 KEY: determine nature of roots given equation 126 ANS: 3 (−5) 2 − 4(2)(0) = 25 TOP: Using the Discriminant PTS: 2 REF: 061423a2 STA: A2.A.2 KEY: determine equation given nature of roots 127 ANS: 2 (−5) 2 − 4(1)(4) = 9 TOP: Using the Discriminant PTS: 2 REF: 011506a2 STA: A2.A.2 128 ANS: 3 3x 2 + x − 14 = 0 1 2 − 4(3)(−14) = 1 + 168 = 169 = 13 2 TOP: Using the Discriminant PTS: 2 REF: 061524a2 STA: A2.A.2 KEY: determine nature of roots given equation TOP: Using the Discriminant 16 ID: A 129 ANS: 3 ± 7 . 2x 2 − 12x + 4 = 0 x 2 − 6x + 2 = 0 x 2 − 6x = −2 x 2 − 6x + 9 = −2 + 9 (x − 3) 2 = 7 x−3= ± 7 x = 3± PTS: 4 130 ANS: 2 x 2 + 2 = 6x 7 REF: fall0936a2 STA: A2.A.24 TOP: Completing the Square STA: A2.A.24 REF: 061122a2 TOP: Completing the Square STA: A2.A.24 STA: A2.A.24 REF: 061408a2 TOP: Completing the Square STA: A2.A.24 STA: A2.A.24 TOP: Completing the Square x 2 − 6x = −2 x 2 − 6x + 9 = −2 + 9 (x − 3) 2 = 7 PTS: 2 REF: 011116a2 131 ANS: 2 PTS: 2 TOP: Completing the Square 132 ANS: 2 (x + 2) 2 = −9 x + 2 =± −9 x = −2 ± 3i PTS: 133 ANS: TOP: 134 ANS: 2 REF: 011408a2 1 PTS: 2 Completing the Square 3 x 2 = 12x − 7 x 2 − 12x = −7 x 2 − 12x + 36 = −7 + 36 (x − 6) 2 = 29 PTS: 2 REF: 061505a2 17 ID: A 135 ANS: 1 2 1 1 − = 1 2 4 64 PTS: 2 136 ANS: 1 y ≥ x2 − x − 6 REF: 081527a2 STA: A2.A.24 TOP: Completing the Square STA: A2.A.4 TOP: Quadratic Inequalities y ≥ (x − 3)(x + 2) PTS: 2 REF: 061017a2 KEY: two variables 137 ANS: 3 or x 2 − 3x − 10 > 0 (x − 5)(x + 2) > 0 x − 5 < 0 and x + 2 < 0 x − 5 > 0 and x + 2 > 0 x < 5 and x < −2 x > 5 and x > −2 x < −2 x>5 PTS: 2 REF: 011115a2 STA: A2.A.4 TOP: Quadratic Inequalities KEY: one variable 138 ANS: x < −1 or x > 5 . x 2 − 4x − 5 > 0. x − 5 > 0 and x + 1 > 0 or x − 5 < 0 and x + 1 < 0 (x − 5)(x + 1) > 0 PTS: 2 KEY: one variable 139 ANS: 2 9 − x2 < 0 x > 5 and x > −1 x < 5 and x < −1 x>5 x < −1 REF: 011228a2 STA: A2.A.4 TOP: Quadratic Inequalities or x + 3 < 0 and x − 3 < 0 x < −3 and x < 3 x2 − 9 > 0 x < −3 (x + 3)(x − 3) > 0 x + 3 > 0 and x − 3 > 0 x > −3 and x > 3 x>3 PTS: 2 KEY: one variable REF: 061507a2 STA: A2.A.4 18 TOP: Quadratic Inequalities ID: A 140 ANS: 2 x 2 − x − 6 = 3x − 6 x 2 − 4x = 0 x(x − 4) = 0 x = 0,4 PTS: 2 REF: 081015a2 STA: A2.A.3 TOP: Quadratic-Linear Systems KEY: equations 141 ANS: 9 1 − , and 1 , 11 . y = x + 5 . 4x 2 + 17x − 4 = x + 5 2 2 2 2 y = 4x 2 + 17x − 4 4x 2 + 16x − 9 = 0 (2x + 9)(2x − 1) = 0 x= − 1 9 and x = 2 2 9 1 1 11 y = − + 5 = and y = + 5 = 2 2 2 2 PTS: 6 KEY: equations 142 ANS: 3 REF: 061139a2 x+y = 5 STA: A2.A.3 TOP: Quadratic-Linear Systems . −5 + y = 5 y = −x + 5 y = 10 (x + 3) 2 + (−x + 5 − 3) 2 = 53 x 2 + 6x + 9 + x 2 − 4x + 4 = 53 2x 2 + 2x − 40 = 0 x 2 + x − 20 = 0 (x + 5)(x − 4) = 0 x = −5,4 PTS: 2 KEY: equations REF: 011302a2 STA: A2.A.3 19 TOP: Quadratic-Linear Systems ID: A 143 ANS: 4 x = 2y . y 2 − (3y) 2 + 32 = 0 . x = 3(−2) = −6 y 2 − 9y 2 = −32 −8y 2 = −32 y2 = 4 y = ±2 PTS: 2 KEY: equations 144 ANS: x(x + 3) = 10 REF: 061312a2 STA: A2.A.3 TOP: Quadratic-Linear Systems x 2 + 3x − 10 = 0 (x + 5)(x − 2) = 0 x = −5, 2 PTS: 2 REF: 011431a2 STA: A2.A.3 TOP: Quadratic-Linear Systems KEY: equations 145 ANS: 2 2 2 2 4 4 2 4 2 2 4 4 x − x + 1. x − 1 = x − 1 x − 1 = x 2 − x − x + 1 = x 2 − x + 1 3 3 9 3 9 3 3 3 3 9 PTS: 2 REF: 081034a2 STA: A2.N.3 TOP: Operations with Polynomials 146 ANS: 2 PTS: 2 REF: 011114a2 STA: A2.N.3 TOP: Operations with Polynomials 147 ANS: 3 3 2 1 37 2 1 37 2 1 1 2 1 6y 3 − y − y. y − y 12y + = 6y 3 + y − 4y 2 − y = 6y 3 − y − y 3 5 10 5 10 5 10 5 2 PTS: 2 REF: 061128a2 148 ANS: 2 The binomials are conjugates, so use FL. STA: A2.N.3 TOP: Operations with Polynomials PTS: 2 REF: 011206a2 149 ANS: 1 The binomials are conjugates, so use FL. STA: A2.N.3 TOP: Operations with Polynomials PTS: 150 ANS: TOP: 151 ANS: TOP: STA: A2.N.3 REF: 011314a2 TOP: Operations with Polynomials STA: A2.N.3 REF: 061407a2 STA: A2.N.3 2 REF: 061201a2 1 PTS: 2 Operations with Polynomials 3 PTS: 2 Operations with Polynomials 20 ID: A 152 ANS: 4 3 x − 1 3 x + 1 − 3 x − 1 = 3 x − 1 (2) = 3x − 2 2 2 2 2 PTS: 153 ANS: TOP: 154 ANS: 3 −2 (−2) −3 2 REF: 011524a2 3 PTS: 2 Operations with Polynomials 3 1 9 8 = = − 9 1 − 8 STA: A2.N.3 REF: 061515a2 TOP: Operations with Polynomials STA: A2.N.3 PTS: 2 REF: 061003a2 STA: A2.N.1 TOP: Negative and Fractional Exponents 155 ANS: 3 6n −1 < 4n −1 . Flip sign when multiplying each side of the inequality by n, since a negative number. 6 4 < n n 6>4 PTS: 2 156 ANS: 4 REF: 061314a2 1 2 f(16) = 4(16) + 16 + 16 = 4(4) + 1 + = 17 PTS: 2 157 ANS: 2 0 − STA: A2.N.1 TOP: Negative and Fractional Exponents STA: A2.N.1 TOP: Negative and Fractional Exponents STA: A2.A.8 REF: 011306a2 TOP: Negative and Fractional Exponents STA: A2.A.8 REF: 011402a2 STA: A2.A.8 REF: 061402a2 STA: A2.A.8 REF: fall0914a2 STA: A2.A.9 1 4 1 2 1 2 REF: 081503a2 1 1 −5 2 w 4 2 −9 = (w ) = w 2 w 158 159 160 161 PTS: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: 2 REF: 081011a2 1 PTS: 2 Negative and Fractional Exponents 1 PTS: 2 Negative and Fractional Exponents 4 PTS: 2 Negative and Fractional Exponents 1 PTS: 2 Negative and Fractional Exponents 21 ID: A 162 ANS: 2 −(x − 1) 1 1−x −1 x x x x −1 1 = = = = − x−1 x−1 x−1 x−1 x −1 PTS: 2 163 ANS: REF: 081018a2 STA: A2.A.9 TOP: Negative Exponents 3y 5 (2x 3 y −7 ) 2 3y 5 (4x 6 y −14 ) 12x 6 y −9 12x 2 3x −4 y 5 12x 2 = = = = 9 . x4 (2x 3 y −7 ) −2 x4 x4 y y9 PTS: 2 164 ANS: 2 REF: 061134a2 STA: A2.A.9 TOP: Negative Exponents STA: A2.A.9 REF: 061210a2 TOP: Negative Exponents STA: A2.A.9 REF: 061324a2 STA: A2.A.9 STA: A2.A.9 REF: 061506a2 TOP: Negative Exponents STA: A2.A.9 STA: A2.A.12 TOP: Evaluating Exponential Expressions REF: 061131a2 STA: A2.A.12 TOP: Evaluating Exponential Expressions REF: 061229a2 STA: A2.A.12 TOP: Evaluating Exponential Expressions 1 1+x +1 x x 1 x −1 + 1 = = = x x+1 x+1 x+1 PTS: 165 ANS: TOP: 166 ANS: TOP: 167 ANS: 2 REF: 011211a2 1 PTS: 2 Negative Exponents 1 PTS: 2 Negative Exponents 2 25b 4 5 2 a −3 b 4 = 3 a PTS: 2 REF: 011514a2 168 ANS: 4 PTS: 2 TOP: Negative Exponents 169 ANS: 2,298.65. PTS: 2 170 ANS: REF: fall0932a2 3 e 3 ln 2 = e ln 2 = e ln 8 = 8 PTS: 2 171 ANS: A = 750e PTS: 2 (0.03)(8) ≈ 953 22 ID: A 172 ANS: 3 4 ⋅5 .03 5000 1 + = 5000(1.0075) 20 ≈ 5805.92 4 PTS: 173 ANS: TOP: 174 ANS: 2 REF: 011410a2 STA: A2.A.12 3 PTS: 2 REF: 061416a2 Evaluating Exponential Expressions 3 p(5) − p(0) = 17(1.15) 2(5) − 17(1.15) 2(0) TOP: Evaluating Exponential Expressions STA: A2.A.12 ≈ 68.8 − 17 ≈ 51 PTS: 2 REF: 061527a2 STA: A2.A.12 175 ANS: 2 4 ⋅ 12 .0325 A = 50 1 + = 50(1.008125) 48 ≈ 73.73 4 TOP: Evaluating Exponential Expressions PTS: 2 176 ANS: 2 8 2 = 64 TOP: Evaluating Exponential Expressions REF: 081511a2 STA: A2.A.12 PTS: 2 REF: fall0909a2 STA: A2.A.18 177 ANS: 4 PTS: 2 REF: 011124a2 TOP: Evaluating Logarithmic Expressions 178 ANS: TOP: Evaluating Logarithmic Expressions STA: A2.A.18 y=0 PTS: 2 REF: 061031a2 STA: A2.A.53 23 TOP: Graphing Exponential Functions ID: A 179 ANS: PTS: 2 REF: 011234a2 180 ANS: 1 PTS: 2 TOP: Graphing Exponential Functions 181 ANS: 2 f −1 (x) = log 4 x STA: A2.A.53 REF: 011506a2 TOP: Graphing Exponential Functions STA: A2.A.53 PTS: 182 ANS: TOP: 183 ANS: TOP: 184 ANS: STA: A2.A.54 REF: 061211a2 TOP: Graphing Logarithmic Functions STA: A2.A.54 REF: 011422a2 STA: A2.A.54 2 REF: fall0916a2 1 PTS: 2 Graphing Logarithmic Functions 3 PTS: 2 Graphing Logarithmic Functions 1 2logx − (3log y + log z) = log x 2 − log y 3 − logz = log PTS: 185 ANS: TOP: 186 ANS: log x 2 2 REF: 061010a2 4 PTS: 2 Properties of Logarithms 2 = log 3a + log 2a x2 y3z STA: A2.A.19 TOP: Properties of Logarithms REF: 061120a2 STA: A2.A.19 KEY: splitting logs 2 logx = log 6a 2 log 6 log a 2 log x = + 2 2 log x = 2 loga 1 log 6 + 2 2 log x = 1 log 6 + log a 2 PTS: KEY: 187 ANS: TOP: 2 REF: 011224a2 splitting logs 4 PTS: 2 Properties of Logarithms STA: A2.A.19 TOP: Properties of Logarithms REF: 061207a2 STA: A2.A.19 KEY: antilogarithms 24 ID: A 188 ANS: 2 log 9 − log 20 log 3 2 − log(10 ⋅ 2) 2 log 3 − (log 10 + log 2) 2b − (1 + a) 2b − a − 1 PTS: 2 REF: 011326a2 KEY: expressing logs algebraically 189 ANS: 3 log4m2 = log 4 + log m2 = log4 + 2 logm STA: A2.A.19 TOP: Properties of Logarithms PTS: 2 REF: 061321a2 KEY: splitting logs 190 ANS: 4 log2x 3 = log 2 + log x 3 = log2 + 3logx STA: A2.A.19 TOP: Properties of Logarithms PTS: 2 REF: 061426a2 KEY: splitting logs 191 ANS: 1 log x = log a 2 + log b STA: A2.A.19 TOP: Properties of Logarithms STA: A2.A.19 TOP: Properties of Logarithms STA: A2.A.28 TOP: Logarithmic Equations log x = log a 2 b x = a2b PTS: 2 REF: 061517a2 KEY: antilogarithms 192 ANS: 4 2 log 4 (5x) = 3 log 4 (5x) = 3 2 5x = 4 3 2 5x = 8 x= 8 5 PTS: 2 KEY: advanced REF: fall0921a2 25 ID: A 193 ANS: 1 x3 + x − 2 =2 x = − ,−1 log x + 3 x 3 x3 + x − 2 = (x + 3) 2 x x3 + x − 2 = x 2 + 6x + 9 x x 3 + x − 2 = x 3 + 6x 2 + 9x 0 = 6x 2 + 8x + 2 0 = 3x 2 + 4x + 1 0 = (3x + 1)(x + 1) 1 x = − ,−1 3 PTS: 6 REF: 081039a2 STA: A2.A.28 KEY: basic 194 ANS: ln(T − T 0 ) = −kt + 4.718 . ln(T − 68) = −0.104(10) + 4.718. TOP: Logarithmic Equations ln(150 − 68) = −k(3) + 4.718 ln(T − 68) = 3.678 T − 68 ≈ 39.6 4.407 ≈ −3k + 4.718 T ≈ 108 k ≈ 0.104 PTS: KEY: 195 ANS: x = 54 6 advanced 3 = 625 PTS: 2 KEY: basic 196 ANS: 800. x = 4 2.5 = 32. y REF: 011139a2 STA: A2.A.28 TOP: Logarithmic Equations REF: 061106a2 STA: A2.A.28 TOP: Logarithmic Equations − 3 2 = 125 y = 125 PTS: 4 KEY: advanced − 2 3 . = 1 25 32 x = = 800 1 y 25 REF: 011237a2 STA: A2.A.28 26 TOP: Logarithmic Equations ID: A 197 ANS: (x + 4) 2 = 17x − 4 x 2 + 8x + 16 = 17x − 4 x 2 − 9x + 20 = 0 (x − 4)(x − 5) = 0 x = 4,5 PTS: 4 KEY: basic 198 ANS: STA: A2.A.28 TOP: Logarithmic Equations PTS: 2 REF: 061329a2 STA: A2.A.28 KEY: advanced 199 ANS: log (x + 3) (2x + 3)(x + 5) = 2 −6 is extraneous TOP: Logarithmic Equations 2x − 1 = 27 REF: 011336a2 4 3 2x − 1 = 81 2x = 82 x = 41 (x + 3) 2 = (2x + 3)(x + 5) x 2 + 6x + 9 = 2x 2 + 13x + 15 x 2 + 7x + 6 = 0 (x + 6)(x + 1) = 0 x = −1 PTS: 6 REF: 011439a2 KEY: applying properties of logarithms 200 ANS: (5x − 1) 1 3 STA: A2.A.28 TOP: Logarithmic Equations STA: A2.A.28 TOP: Logarithmic Equations REF: 011503a2 KEY: basic STA: A2.A.28 =4 5x − 1 = 64 5x = 65 x = 13 PTS: KEY: 201 ANS: TOP: 2 REF: 061433a2 advanced 3 PTS: 2 Logarithmic Equations 27 ID: A 202 ANS: (x + 1) 3 = 64 x+1 = 4 x=3 PTS: 2 REF: 061531a2 KEY: basic 203 ANS: 2 23 ± x − 7x + 12 = 3 log 2 x = 2x − 10 STA: A2.A.28 TOP: Logarithmic Equations (−23) 2 − 4(1)(92) ≈ 17.84, 5.16 2(1) x 2 − 7x + 12 =8 2x − 10 x 2 − 7x + 12 = 16x − 80 x 2 − 23x + 92 = 0 PTS: 6 REF: 081539a2 KEY: applying properties of logarithms 204 ANS: 3 75000 = 25000e .0475t STA: A2.A.28 TOP: Logarithmic Equations STA: A2.A.6 TOP: Exponential Growth 3 = e .0475t ln 3 = ln e .0475t ln 3 .0475t ⋅ ln e = .0475 .0475 23.1 ≈ t PTS: 2 REF: 061117a2 28 ID: A 205 ANS: 2 320 = 10(2) 32 = (2) t 60 log 32 = log(2) log 32 = t 60 t 60 t log 2 60 60 log 32 =t log 2 300 = t PTS: 2 206 ANS: 30700 = 50e 3t REF: 011205a2 STA: A2.A.6 TOP: Exponential Growth REF: 011333a2 STA: A2.A.6 TOP: Exponential Growth REF: 061313a2 STA: A2.A.6 TOP: Exponential Growth 614 = e 3t ln 614 = ln e 3t ln 614 = 3t ln e ln 614 = 3t 2.14 ≈ t PTS: 2 207 ANS: 3 1000 = 500e .05t 2 = e .05t ln 2 = ln e .05t ln 2 .05t ⋅ ln e = .05 .05 13.9 ≈ t PTS: 2 29 ID: A 208 ANS: 3 4x (2 2 ) 2 2 x + 4x 2 2 + 4x 2x + 8x = 2 −6 . 2x 2 + 8x = −6 2x 2 + 8x + 6 = 0 = 2 −6 =2 x 2 + 4x + 3 = 0 −6 (x + 3)(x + 1) = 0 x = −3 x = −1 PTS: 2 REF: 061015a2 KEY: common base shown 209 ANS: 4 9 3x + 1 = 27 x + 2 . STA: A2.A.27 TOP: Exponential Equations STA: A2.A.27 TOP: Exponential Equations STA: A2.A.27 TOP: Exponential Equations (3 2 ) 3x + 1 = (3 3 ) x + 2 3 6x + 2 = 3 3x + 6 6x + 2 = 3x + 6 3x = 4 x= 4 3 PTS: 2 REF: 081008a2 KEY: common base not shown 210 ANS: 16 2x + 3 = 64 x + 2 (4 2 ) 2x + 3 = (4 3 ) x + 2 4x + 6 = 3x + 6 x=0 PTS: 2 REF: 011128a2 KEY: common base not shown 30 ID: A 211 ANS: 2 4 2x + 5 = 8 3x . 2 2x + 5 3 3x 2 = 2 2 4x + 10 = 2 9x 4x + 10 = 9x 10 = 5x 2=x PTS: 2 REF: 061105a2 KEY: common base not shown 212 ANS: 3 81 x + 2x 4 x 3 3 4x 2 = 27 STA: A2.A.27 TOP: Exponential Equations STA: A2.A.27 TOP: Exponential Equations STA: A2.A.27 TOP: Exponential Equations 5x 3 3 + 2x 2 = 3 3 3 + 8x 2 = 3 5x 5x 3 4x 3 + 8x 2 − 5x = 0 x(4x 2 + 8x − 5) = 0 x(2x − 1)(2x + 5) = 0 x = 0, 5 1 ,− 2 2 PTS: 6 REF: 061239a2 KEY: common base not shown 213 ANS: 4 8 3k + 4 = 4 2k − 1 . (2 3 ) 3k + 4 = (2 2 ) 2k − 1 2 9k + 12 = 2 4k − 2 9k + 12 = 4k − 2 5k = −14 k= − 14 5 PTS: 2 REF: 011309a2 KEY: common base not shown 31 ID: A 214 ANS: ln e 4x = ln 12 4x = ln 12 x= ln 12 4 ≈ 0.62 PTS: 2 REF: 011530a2 KEY: without common base 215 ANS: x−1 5 4x = 5 3 STA: A2.A.27 TOP: Exponential Equations STA: A2.A.27 TOP: Exponential Equations STA: A2.A.27 TOP: Exponential Equations 4x = 3x − 3 x = −3 PTS: 2 REF: 061528a2 KEY: common base shown 216 ANS: 2 −4 = 2 3x − 1 −4 = 3x − 1 −3 = 3x −1 = x PTS: 2 REF: 081529a2 KEY: common base shown 217 ANS: 1 2 3 2 2 5 C 3 (3x) (−2) = 10 ⋅ 9x ⋅ −8 = −720x PTS: 2 REF: fall0919a2 STA: A2.A.36 TOP: Binomial Expansions 218 ANS: 32x 5 − 80x 4 + 80x 3 − 40x 2 + 10x − 1. 5 C 0 (2x) 5 (−1) 0 = 32x 5 . 5 C 1 (2x) 4 (−1) 1 = −80x 4 . 5 C 2 (2x) 3 (−1) 2 = 80x 3 . 5 C 3 (2x) 2 (−1) 3 = −40x 2 . 5 C 4 (2x) 1 (−1) 4 = 10x. 5 C 5 (2x) 0 (−1) 5 = −1 PTS: 4 REF: 011136a2 219 ANS: 1 6 3 6 3 9 C 3 a (−4b) = −5376a b STA: A2.A.36 TOP: Binomial Expansions PTS: 2 REF: 061126a2 220 ANS: 3 4 1 2 4 2 3 C 2 (2x ) (−y) = 6x y STA: A2.A.36 TOP: Binomial Expansions STA: A2.A.36 TOP: Binomial Expansions PTS: 2 REF: 011215a2 32 ID: A 221 ANS: 3 3 x 3 (−2y) 3 = 20 ⋅ x ⋅ −8y 3 = −20x 3 y 3 C 6 3 2 8 PTS: 2 REF: 061215a2 222 ANS: 3 8−3 ⋅ (−2) 3 = 56x 5 ⋅ (−8) = −448x 5 8 C3 ⋅ x STA: A2.A.36 TOP: Binomial Expansions PTS: 2 REF: 011308a2 223 ANS: 1 5−2 (−3) 2 = 720x 3 5 C 2 (2x) STA: A2.A.36 TOP: Binomial Expansions PTS: 224 ANS: TOP: 225 ANS: 3 ± ,− 2 STA: A2.A.36 REF: 081525a2 TOP: Binomial Expansions STA: A2.A.36 2 REF: 011519a2 3 PTS: 2 Binomial Expansions 1 . 2 8x 3 + 4x 2 − 18x − 9 = 0 4x 2 (2x + 1) − 9(2x + 1) = 0 (4x 2 − 9)(2x + 1) = 0 4x 2 − 9 = 0 or 2x + 1 = 0 (2x + 3)(2x − 3) = 0 x= ± PTS: 4 226 ANS: 2 x 3 + x 2 − 2x = 0 x= − 1 2 3 2 REF: fall0937a2 STA: A2.A.26 TOP: Solving Polynomial Equations REF: 011103a2 STA: A2.A.26 TOP: Solving Polynomial Equations x(x 2 + x − 2) = 0 x(x + 2)(x − 1) = 0 x = 0,−2,1 PTS: 2 33 ID: A 227 ANS: 3 3x 5 − 48x = 0 3x(x 4 − 16) = 0 3x(x 2 + 4)(x 2 − 4) = 0 3x(x 2 + 4)(x + 2)(x − 2) = 0 PTS: 2 REF: 011216a2 228 ANS: x 4 + 4x 3 + 4x 2 + 16x = 0 STA: A2.A.26 TOP: Solving Polynomial Equations STA: A2.A.26 TOP: Solving Polynomial Equations STA: A2.A.26 TOP: Solving Polynomial Equations STA: A2.A.26 REF: 061005a2 TOP: Solving Polynomial Equations STA: A2.A.50 STA: A2.A.50 TOP: Solving Polynomial Equations x(x 3 + 4x 2 + 4x + 16) = 0 x(x 2 (x + 4) + 4(x + 4)) = 0 x(x 2 + 4)(x + 4) = 0 x = 0,±2i,−4 PTS: 6 REF: 061339a2 229 ANS: x 3 + 5x 2 − 4x − 20 = 0 x 2 (x + 5) − 4(x + 5) = 0 (x 2 − 4)(x + 5) = 0 (x + 2)(x − 2)(x + 5) = 0 x = ±2,−5 PTS: 4 REF: 061437a2 230 ANS: x 2 (2x − 1) − 4(2x − 1) = 0 (x 2 − 4)(2x − 1) = 0 (x + 2)(x − 2)(2x − 1) = 0 x = ± 2, 1 2 PTS: 4 REF: 081537a2 231 ANS: 4 PTS: 2 TOP: Solving Polynomial Equations 232 ANS: 2 The roots are −1,2,3. PTS: 2 REF: 081023a2 34 ID: A 233 ANS: 4 PTS: 234 ANS: TOP: 235 ANS: (3 + STA: A2.A.50 REF: 081501a2 TOP: Solving Polynomial Equations STA: A2.A.50 PTS: 2 REF: 081001a2 KEY: without variables | index = 2 236 ANS: a2b3 − 4 STA: A2.N.4 TOP: Operations with Irrational Expressions PTS: 2 KEY: index > 2 237 ANS: 3 REF: 011231a2 STA: A2.A.13 TOP: Simplifying Radicals REF: 061204a2 STA: A2.A.13 TOP: Simplifying Radicals 3 2 REF: 061222a2 1 PTS: 2 Solving Polynomial Equations 4 5)(3 − 5) = 9 − 25 = 4 4 3 a 15 a = 4a 5 3 a PTS: 2 KEY: index > 2 238 ANS: 5 3x 3 − 2 27x 3 = 5 x 2 PTS: 2 239 ANS: 3 3 3x − 2 9x 2 REF: 061032a2 3x = 5x 3x − 6x STA: A2.N.2 3x = −x 3x TOP: Operations with Radicals 6a 4 b 2 + 3 (27 ⋅ 6)a 4 b 2 a 3 6ab 2 + 3a 4a 3 3 6ab 2 6ab 2 PTS: 2 REF: 011319a2 STA: A2.N.2 240 ANS: 4 3 27x 2 3 16x 4 = 3 3 3 ⋅ 2 4 ⋅ x 6 = 3 ⋅ 2 ⋅ x 2 3 2 = 6x 2 3 2 PTS: 2 REF: 011421a2 STA: A2.N.2 35 TOP: Operations with Radicals TOP: Operations with Radicals ID: A 241 ANS: 1 3 64a 5 b 6 = 3 4 3 a 3 a 2 b 6 = 4ab 2 PTS: 2 242 ANS: 4 a2 REF: 011516a2 2b − 3a 4ab 3 2b + 7ab 9b 2 6b = 4ab PTS: 2 REF: fall0918a2 KEY: with variables | index = 2 243 ANS: 108x 5 y 8 = STA: A2.N.2 18x 4 y 3 = 3x 2 y 2b − 9ab TOP: Operations with Radicals 2b + 7ab 6b = −5ab 2b + 7ab 6b STA: A2.A.14 TOP: Operations with Radicals STA: A2.A.14 TOP: Operations with Radicals STA: A2.A.14 TOP: Operations with Radicals 2y 6xy 5 PTS: 2 REF: 011133a2 KEY: with variables | index = 2 244 ANS: 1 3 27a 3 ⋅ 4 16b 8 = 3a ⋅ 2b 2 = 6ab 2 PTS: 2 REF: 061504a2 KEY: with variables | index > 2 245 ANS: 5(3 + 2) 3+ 5 × . 7 3− 2 3+ PTS: 2 246 ANS: 1 3 +5 3 −5 5− 3 +5 ⋅ 3 +5 = 5(3 + 2) 5(3 + 2) = 9−2 7 13 13 5 + 13 ⋅ STA: A2.N.5 TOP: Rationalizing Denominators 3 + 5 3 + 5 3 + 25 28 + 10 3 14 + 5 3 = = − 3 − 25 −22 11 REF: 061012a2 5+ PTS: 2 2 = REF: fall0928a2 PTS: 2 247 ANS: 3 4 2 = STA: A2.N.5 TOP: Rationalizing Denominators 4(5 + 13) 5 + 13 = 25 − 13 3 REF: 061116a2 STA: A2.N.5 36 TOP: Rationalizing Denominators ID: A Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic Answer Section 248 ANS: 1 7+ 11 11 7 + 11 1 ⋅ 7− = PTS: 2 249 ANS: 1 REF: 011404a2 4+ 11 11 4 + 11 5 ⋅ 4− = PTS: 2 250 ANS: 4 3− 8 3 ⋅ 3 3 = 3 3− 3 = 3a b 3b 3 ⋅ a 3b 3b PTS: 2 KEY: index = 2 252 ANS: 4 2x + 4 ⋅ x+2 x− PTS: KEY: 254 ANS: TOP: x = × = x+2 x+2 = x x+ x 11 3− 2 3 STA: A2.N.5 TOP: Rationalizing Denominators 6 TOP: Rationalizing Denominators 3 3b 3b = 3ab ab STA: A2.A.15 TOP: Rationalizing Denominators 2(x + 2) x + 2 = 2 x+2 x+2 REF: 011122a2 x+ TOP: Rationalizing Denominators STA: A2.N.5 3 3 −2 6 = 3 REF: 081019a2 PTS: 2 KEY: index = 2 253 ANS: 4 x 24 REF: 081518a2 2 STA: A2.N.5 5(4 + 11) 5(4 + 11) = = 4+ 16 − 11 5 REF: 061509a2 PTS: 2 251 ANS: 3 3 7 + 11 7 + 11 = 49 − 11 38 = STA: A2.A.15 TOP: Rationalizing Denominators x(x + x ) x + x x2 + x x = = 2 x(x − 1) x−1 x −x 2 REF: 061325a2 index = 2 1 PTS: 2 Solving Radicals STA: A2.A.15 TOP: Rationalizing Denominators REF: 061018a2 STA: A2.A.22 KEY: extraneous solutions 1 ID: A 255 ANS: 3 3x + 16 = (x + 2) 2 . −4 is an extraneous solution. 3x + 16 = x 2 + 4x + 4 0 = x 2 + x − 12 0 = (x + 4)(x − 3) x = −4 x = 3 PTS: 2 REF: 061121a2 KEY: extraneous solutions 256 ANS: 7. 4 − 2x − 5 = 1 STA: A2.A.22 TOP: Solving Radicals STA: A2.A.22 TOP: Solving Radicals − 2x − 5 = −3 2x − 5 = 9 2x = 14 x=7 PTS: 2 KEY: basic 257 ANS: 1 5x + 29 = (x + 3) 2 REF: 011229a2 . (−5) + 3 shows an extraneous solution. 5x + 29 = x 2 + 6x + 9 0 = x 2 + x − 20 0 = (x + 5)(x − 4) x = −5,4 PTS: 2 REF: 061213a2 KEY: extraneous solutions STA: A2.A.22 2 TOP: Solving Radicals ID: A 258 ANS: x 2 + x − 1 = −4x + 3 x 2 + x − 1 = 16x 2 − 24x + 9 2 −4 + 3 ≥ 0 3 1 ≥0 3 0 = 15x 2 − 25x + 10 −4(1) + 3 < 0 0 = 3x 2 − 5x + 2 1 is extraneous 0 = (3x − 2)(x − 1) x= 2 ,x≠1 3 PTS: 6 REF: 011339a2 KEY: extraneous solutions 259 ANS: 2 2x − 4 = x − 2 STA: A2.A.22 TOP: Solving Radicals STA: A2.A.22 TOP: Solving Radicals REF: 061011a2 STA: A2.A.10 STA: A2.A.10 TOP: Fractional Exponents as Radicals STA: A2.A.11 TOP: Radicals as Fractional Exponents STA: A2.A.11 TOP: Radicals as Fractional Exponents 2x − 4 = x 2 − 4x + 4 0 = x 2 − 6x + 8 0 = (x − 4)(x − 2) x = 4,2 PTS: KEY: 260 ANS: TOP: 261 ANS: − x 2 5 2 REF: 061406a2 extraneous solutions 2 PTS: 2 Fractional Exponents as Radicals 4 1 = x 2 5 = 1 5 x2 PTS: 2 262 ANS: 1 4 REF: 011118a2 1 4 2 4 16x y = 16 x y 2 7 PTS: 2 263 ANS: 1 4 PTS: 2 1 2 = 2x y 7 4 REF: 061107a2 1 4 2 4 81x y = 81 x y 2 7 4 5 5 4 1 2 = 3x y 5 4 REF: 081504a2 3 ID: A 264 ANS: 3 −300 = PTS: 2 265 ANS: 3 9 −1 100 −1 3 REF: 061006a2 2− 16 PTS: 2 266 ANS: 4 −1 STA: A2.N.6 TOP: Square Roots of Negative Numbers 2 = 3i 2 − 4i 2 = −i 2 REF: 061404a2 STA: A2.N.6 TOP: Square Roots of Negative Numbers PTS: 2 REF: 081524a2 267 ANS: 1 PTS: 2 TOP: Imaginary Numbers 268 ANS: 1 2i 2 + 3i 3 = 2(−1) + 3(−i) = −2 − 3i STA: A2.N.6 REF: 061019a2 TOP: Square Roots of Negative Numbers STA: A2.N.7 PTS: 2 REF: 081004a2 269 ANS: i 13 + i 18 + i 31 + n = 0 STA: A2.N.7 TOP: Imaginary Numbers PTS: 2 REF: 061228a2 STA: A2.N.7 270 ANS: 4xi + 5yi 8 + 6xi 3 + 2yi 4 = 4xi + 5y − 6xi + 2y = 7y − 2xi TOP: Imaginary Numbers PTS: 2 REF: 011433a2 271 ANS: xi 8 − yi 6 = x(1) − y(−1) = x + y STA: A2.N.7 TOP: Imaginary Numbers STA: A2.N.7 REF: 081024a2 TOP: Imaginary Numbers STA: A2.N.8 REF: 011111a2 STA: A2.N.8 REF: 011213a2 STA: A2.N.8 REF: 061219a2 STA: A2.N.8 −180x 16 = 6x 8 i 5 i + (−1) − i + n = 0 −1 + n = 0 n=1 272 273 274 275 PTS: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: 2 REF: 061533a2 2 PTS: 2 Conjugates of Complex Numbers 4 PTS: 2 Conjugates of Complex Numbers 2 PTS: 2 Conjugates of Complex Numbers 3 PTS: 2 Conjugates of Complex Numbers 4 ID: A 276 ANS: 2 (3 − 7i)(3 − 7i) = 9 − 21i − 21i + 49i 2 = 9 − 42i − 49 = −40 − 42i PTS: 2 REF: fall0901a2 STA: A2.N.9 TOP: Multiplication and Division of Complex Numbers 277 ANS: 4 (x + i) 2 − (x − i) 2 = x 2 + 2xi + i 2 − (x 2 − 2xi + i 2 ) = 4xi PTS: 2 REF: 011327a2 STA: A2.N.9 TOP: Multiplication and Division of Complex Numbers 278 ANS: 3 (3i)(2i) 2 (m + i) (3i)(4i 2 )(m + i) (3i)(−4)(m + i) (−12i)(m + i) −12mi − 12i 2 −12mi + 12 PTS: 2 REF: 061319a2 STA: A2.N.9 TOP: Multiplication and Division of Complex Numbers 279 ANS: (x + yi)(x − yi) = x 2 − y 2 i 2 = x 2 + y 2 PTS: 2 REF: 061432a2 STA: A2.N.9 TOP: Multiplication and Division of Complex Numbers 280 ANS: 4 (−3 − 2i)(−3 + 2i) = 9 − 4i 2 = 9 + 4 = 13 PTS: 2 REF: 011512a2 STA: A2.N.9 TOP: Multiplication and Division of Complex Numbers 281 ANS: 2xi(i − 4i 2 ) = 2xi 2 − 8xi 3 = 2xi 2 − 8xi 3 = −2x + 8xi PTS: 2 REF: 011533a2 STA: A2.N.9 TOP: Multiplication and Division of Complex Numbers 5 ID: A 282 ANS: −2(x 2 + 6) x4 . x 2 (x − 3) + 6(x − 3) x 2 − 4x ⋅ 2x − 4 x 2 + 2x − 8 ÷ x 4 − 3x 3 16 − x 2 (x 2 + 6)(x − 3) 2(x − 2) (4 + x)(4 − x) ⋅ 3 ⋅ x(x − 4) x (x − 3) (x + 4)(x − 2) −2(x 2 + 6) x4 PTS: 6 REF: 011239a2 STA: A2.A.16 KEY: division 283 ANS: −(x 2 − 4) −(x + 2)(x − 2) −(x + 2) x+3 1 × = × = (x + 4)(x + 3) 2(x − 2) x+4 2(x − 2) 2(x + 4) TOP: Multiplication and Division of Rationals PTS: 4 REF: 061236a2 STA: A2.A.16 KEY: division 284 ANS: 4 (x + 11)(x − 2) −1 −1 x 2 + 9x − 22 ÷ (2 − x) = ⋅ = 2 (x + 11)(x − 11) x − 2 x − 11 x − 121 TOP: Multiplication and Division of Rationals PTS: 2 REF: 011423a2 KEY: Division 285 ANS: (6 − x)(6 + x) (x + 6)(x − 3) ⋅ = 6−x x−3 (x + 6)(x + 6) STA: A2.A.16 TOP: Multiplication and Division of Rationals 2 REF: 011529a2 STA: A2.A.16 division 3 3y 3y − 9 3(y − 3) 3 9 9 + = − = = = 6 − 2y 2y − 6 2y − 6 2y − 6 2(y − 3) 2 TOP: Multiplication and Division of Rationals PTS: KEY: 286 ANS: 3y 2y − 6 PTS: 2 REF: 011325a2 STA: A2.A.16 6 TOP: Addition and Subtraction of Rationals ID: A 287 ANS: no solution. 4x 12 = 2+ x−3 x−3 4x − 12 =2 x−3 4(x − 3) =2 x−3 4≠ 2 PTS: 2 REF: fall0930a2 KEY: rational solutions 288 ANS: 1 1 2 4 − = x + 3 3 − x x2 − 9 3 STA: A2.A.23 TOP: Solving Rationals STA: A2.A.23 TOP: Solving Rationals 2 4 1 + = x + 3 x − 3 x2 − 9 x − 3 + 2(x + 3) 4 = (x + 3)(x − 3) (x + 3)(x − 3) x − 3 + 2x + 6 = 4 3x = 1 x= 1 3 PTS: 4 REF: 081036a2 KEY: rational solutions 289 ANS: 13 = 10 − x x . x= 10 ± 100 − 4(1)(13) 10 ± 48 10 ± 4 3 = = = 5±2 3 2(1) 2 2 13 = 10x − x 2 x 2 − 10x + 13 = 0 PTS: 4 REF: 061336a2 KEY: irrational and complex solutions STA: A2.A.23 7 TOP: Solving Rationals ID: A 290 ANS: 2 (x + 3)(x − 3) 5(x + 3) 30 + = 3 is an extraneous root. (x + 3)(x − 3) (x + 3)(x − 3) (x − 3)(x + 3) 30 + x 2 − 9 = 5x + 15 x 2 − 5x + 6 = 0 (x − 3)(x − 2) = 0 x=2 PTS: 2 REF: 061417a2 KEY: rational solutions 291 ANS: 3 2(x − 3) x(x − 3) 5x − = x(x − 3) x(x − 3) x(x − 3) STA: A2.A.23 TOP: Solving Rationals STA: A2.A.23 TOP: Solving Rationals STA: A2.A.23 TOP: Solving Rationals 5x − 2x + 6 = x 2 − 3x 0 = x 2 − 6x − 6 PTS: 2 REF: 011522a2 KEY: irrational and complex solutions 292 ANS: x 2 3 + = − x+2 x x+2 x+2 3 = − x+2 x 1= − 3 x x = −3 PTS: 4 REF: 061537a2 KEY: rational solutions 8 ID: A 293 ANS: 10x 1 x = + x 4 4 9x 1 = 4 x 9x 2 = 4 x2 = 4 9 x= ± 2 3 PTS: KEY: 294 ANS: x + 16 x−2 2 REF: 081534a2 STA: A2.A.23 TOP: Solving Rationals rational solutions 3 7(x − 2) − ≤ 0 −6x + 30 = 0 x − 2 = 0 . Check points such that x < 2, 2 < x < 5, and x > 5. If x = 1 , x−2 x=2 −6x = −30 −6x + 30 ≤0 x−2 x=5 −6(1) + 30 24 −6(3) + 30 12 = = −24, which is less than 0. If x = 3, = = 12, which is greater than 0. If x = 6 , −1 1 1−2 3−2 −6(6) + 30 −6 3 = = − , which is less than 0. 6−2 4 2 PTS: 2 295 ANS: 2 REF: 011424a2 STA: A2.A.23 TOP: Rational Inequalities x 1 x2 − 4 − (x + 2)(x − 2) 4 x 4x 8x = = × = x−2 2(x + 2) 1 1 2x + 4 4x + 2x 4 8x PTS: 2 REF: fall0920a2 STA: A2.A.17 296 ANS: d−8 1 4 − 2d 2 d d − 8 2d 2 d − 8 = = × = 3 1 2d + 3d 5d 2d 5 + 2 d 2d 2d PTS: 2 REF: 061035a2 STA: A2.A.17 9 TOP: Complex Fractions TOP: Complex Fractions ID: A 297 ANS: 2 4 1− x(x − 4) x x2 x 2 − 4x x × 2 = 2 = = 2 8 x − 2x − 8 (x − 4)(x + 2) x + 2 x 1− − 2 x x PTS: 2 REF: 061305a2 STA: A2.A.17 298 ANS: 3 b ac + b a+ c c ac + b c ac + b = = ⋅ = c cd − b cd − b b cd − b d− c c TOP: Complex Fractions PTS: 2 REF: 011405a2 STA: A2.A.17 299 ANS: 3 1+ x(x + 3) x x2 x 2 + 3x x ⋅ 2 = 2 = = 5 24 x − 5x − 24 (x − 8)(x + 3) x − 8 1− − 2 x x x TOP: Complex Fractions PTS: 4 300 ANS: 12 ⋅ 6 = 9w REF: 061436a2 STA: A2.A.17 TOP: Complex Fractions REF: 011130a2 STA: A2.A.5 TOP: Inverse Variation REF: 011226a2 STA: A2.A.5 TOP: Inverse Variation 8=w PTS: 2 301 ANS: 1 3 3 10 ⋅ = p 2 5 15 = 3 p 5 25 = p PTS: 2 10 ID: A 302 ANS: 1 20(−2) = x(−2x + 2) −40 = −2x 2 + 2x 2x 2 − 2x − 40 = 0 x 2 − x − 20 = 0 (x + 4)(x − 5) = 0 x = −4,5 PTS: 2 REF: 011321a2 303 ANS: 2 2 2 ⋅ 3 = 12 . 6 2 d = 12 42 ⋅ STA: A2.A.5 TOP: Inverse Variation REF: 061310a2 STA: A2.A.5 TOP: Inverse Variation REF: 011412a2 STA: A2.A.5 TOP: Inverse Variation STA: A2.A.5 REF: 061510a2 TOP: Inverse Variation STA: A2.A.5 REF: 081507a2 STA: A2.A.5 TOP: Inverse Variation REF: fall0927a2 STA: A2.A.40 TOP: Functional Notation 36d = 12 3 = 12 4 1 d= 3 PTS: 2 304 ANS: 3 20 ⋅ 2 = −5t −8 = t PTS: 2 305 ANS: 25 ⋅ 6 = 30q 5=q PTS: 2 REF: 011528a2 306 ANS: 2 PTS: 2 TOP: Inverse Variation 307 ANS: 4 3 ⋅ 400 = 8x 150 = x PTS: 2 308 ANS: 4 y − 2 sin θ = 3 y = 2 sin θ + 3 f(θ) = 2 sin θ + 3 PTS: 2 11 ID: A 309 ANS: 2 f 10 = −10 −10 5 = =− 2 42 (−10) − 16 84 PTS: 2 310 ANS: g(10) = a(10) REF: 061102a2 STA: A2.A.41 TOP: Functional Notation 2 1 − 10 = 100a 2 (−9) = −900a 2 PTS: 2 REF: 061333a2 311 ANS: 4 f a + 1 = 4(a + 1) 2 − (a + 1) + 1 STA: A2.A.41 TOP: Functional Notation = 4(a 2 + 2a + 1) − a = 4a 2 + 8a + 4 − a = 4a 2 + 7a + 4 PTS: 2 REF: 011527a2 STA: A2.A.41 TOP: Functional Notation 312 ANS: 3 PTS: 2 REF: 011119a2 STA: A2.A.52 TOP: Families of Functions 313 ANS: 4 PTS: 2 REF: 011219a2 STA: A2.A.52 TOP: Properties of Graphs of Functions and Relations 314 ANS: 3 As originally written, alternatives (2) and (3) had no domain restriction, so that both were correct. 315 316 317 318 319 320 321 322 323 PTS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: 2 REF: 061405a2 STA: A2.A.52 Properties of Graphs of Functions and Relations 1 PTS: 2 REF: 061004a2 Identifying the Equation of a Graph 2 PTS: 2 REF: 061108a2 Identifying the Equation of a Graph 2 PTS: 2 REF: 011301a2 Identifying the Equation of a Graph 2 PTS: 2 REF: 011502a2 Identifying the Equation of a Graph 3 PTS: 2 REF: 011305a2 Defining Functions 4 PTS: 2 REF: fall0908a2 Defining Functions KEY: graphs 1 PTS: 2 REF: 061013a2 Defining Functions 4 PTS: 2 REF: 011101a2 Defining Functions KEY: graphs 3 PTS: 2 REF: 061114a2 Defining Functions KEY: graphs 12 STA: A2.A.52 STA: A2.A.52 STA: A2.A.52 STA: A2.A.52 STA: A2.A.37 STA: A2.A.38 STA: A2.A.38 STA: A2.A.38 STA: A2.A.38 ID: A 324 ANS: 1 PTS: 2 REF: 061409a2 STA: A2.A.38 TOP: Defining Functions KEY: graphs 325 ANS: 2 PTS: 2 REF: 011507a2 STA: A2.A.38 TOP: Defining Functions KEY: graphs 326 ANS: 4 (4) fails the horizontal line test. Not every element of the range corresponds to only one element of the domain. PTS: 2 REF: fall0906a2 STA: A2.A.43 TOP: Defining Functions 327 ANS: 3 (1) and (4) fail the horizontal line test and are not one-to-one. Not every element of the range corresponds to only one element of the domain. (2) fails the vertical line test and is not a function. Not every element of the domain corresponds to only one element of the range. PTS: 2 REF: 081020a2 ANS: 2 PTS: 2 TOP: Defining Functions ANS: 2 PTS: 2 TOP: Defining Functions ANS: 4 PTS: 2 TOP: Defining Functions ANS: 2 PTS: 2 TOP: Defining Functions ANS: 3 PTS: 2 TOP: Defining Functions ANS: 3 PTS: 2 TOP: Domain and Range ANS: 4 PTS: 2 TOP: Domain and Range ANS: 2 PTS: 2 TOP: Domain and Range ANS: 1 PTS: 2 TOP: Domain and Range ANS: 1 PTS: 2 TOP: Domain and Range ANS: 2 PTS: 2 TOP: Domain and Range ANS: 3 PTS: 2 TOP: Domain and Range ANS: 2 PTS: 2 TOP: Domain and Range ANS: D: −5 ≤ x ≤ 8. R: −3 ≤ y ≤ 2 STA: A2.A.43 REF: 011225a2 TOP: Defining Functions STA: A2.A.43 REF: 061218a2 STA: A2.A.43 REF: 061303a2 STA: A2.A.43 REF: 011407a2 STA: A2.A.43 REF: 061501a2 STA: A2.A.43 REF: KEY: REF: KEY: REF: KEY: REF: KEY: REF: KEY: REF: KEY: REF: KEY: REF: STA: A2.A.39 PTS: 2 REF: 011132a2 342 ANS: 1 PTS: 2 TOP: Domain and Range STA: A2.A.51 REF: 061202a2 328 329 330 331 332 333 334 335 336 337 338 339 340 341 fall0923a2 real domain 061112a2 real domain 011222a2 real domain 011313a2 real domain 011416a2 real domain 011521a2 real domain 081517a2 real domain 081003a2 13 STA: A2.A.39 STA: A2.A.39 STA: A2.A.39 STA: A2.A.39 STA: A2.A.39 STA: A2.A.39 STA: A2.A.51 TOP: Domain and Range STA: A2.A.51 ID: A 343 ANS: TOP: 344 ANS: TOP: 345 ANS: TOP: 346 ANS: 3 PTS: 2 REF: 061308ge Domain and Range 3 PTS: 2 REF: 061418a2 Domain and Range 4 PTS: 2 REF: 061518a2 Domain and Range 3 1 f(4) = (4) − 3 = −1. g(−1) = 2(−1) + 5 = 3 2 PTS: 2 REF: fall0902a2 KEY: numbers 347 ANS: 2 6(x 2 − 5) = 6x 2 − 30 STA: A2.A.51 STA: A2.A.51 STA: A2.A.51 STA: A2.A.42 TOP: Compositions of Functions PTS: 2 REF: 011109a2 STA: A2.A.42 KEY: variables 348 ANS: 7. f(−3) = (−3) 2 − 6 = 3. g(x) = 2 3 − 1 = 7. TOP: Compositions of Functions PTS: 2 REF: 061135a2 KEY: numbers 349 ANS: 4 1 1 g = = 2. f(2) = 4(2) − 2 2 = 4 2 1 2 TOP: Compositions of Functions STA: A2.A.42 PTS: KEY: 350 ANS: TOP: 351 ANS: 2 REF: 011204a2 STA: A2.A.42 TOP: Compositions of Functions numbers 2 PTS: 2 REF: 061216a2 STA: A2.A.42 Compositions of Functions KEY: variables 3 1 1 h(−8) = (−8) − 2 = −4 − 2 = −6. g(−6) = (−6) + 8 = −3 + 8 = 5 2 2 PTS: 2 REF: 011403a2 STA: A2.A.42 TOP: Compositions of Functions KEY: numbers 352 ANS: 1 f(g(x)) = 2(x + 5) 2 − 3(x + 5) + 1 = 2(x 2 + 10x + 25) − 3x − 15 + 1 = 2x 2 + 17x + 36 PTS: 2 KEY: variables REF: 061419a2 STA: A2.A.42 14 TOP: Compositions of Functions ID: A 353 ANS: 4 g(−2) = 3(−2) − 2 = −8 f(−8) = 2(−8) 2 + 1 = 128 + 1 = 129 PTS: 2 REF: 061503a2 STA: A2.A.42 KEY: numbers 354 ANS: (x + 1) 2 − (x + 1) = x 2 + 2x + 1 − x − 1 = x 2 + x TOP: Compositions of Functions PTS: KEY: 355 ANS: TOP: 356 ANS: STA: A2.A.42 TOP: Compositions of Functions REF: 081027a2 KEY: equations STA: A2.A.44 2 REF: 081530a2 variables 3 PTS: 2 Inverse of Functions y = x 2 − 6 . f −1 (x) is not a function. x = y2 − 6 x + 6 = y2 ± x+6 = y 357 358 359 360 361 PTS: KEY: ANS: TOP: ANS: TOP: ANS: TOP: ANS: TOP: ANS: PTS: TOP: 362 ANS: TOP: 2 REF: 061132a2 STA: A2.A.44 equations 2 PTS: 2 REF: 061521a2 Inverse of Functions KEY: equations 2 PTS: 2 REF: 081523a2 Inverse of Functions KEY: ordered pairs 2 PTS: 2 REF: fall0926a2 Transformations with Functions and Relations 1 PTS: 2 REF: 081022a2 Transformations with Functions and Relations 2 REF: 061435a2 STA: A2.A.46 Transformations with Functions and Relations 1 PTS: 2 REF: 061516a2 Transformations with Functions and Relations 15 TOP: Inverse of Functions STA: A2.A.44 STA: A2.A.44 STA: A2.A.46 STA: A2.A.46 STA: A2.A.46 ID: A 363 ANS: 4 PTS: 2 TOP: Sequences 364 ANS: 1 common difference is 2. b n = x + 2n REF: 061026a2 STA: A2.A.29 STA: A2.A.29 TOP: Sequences PTS: 2 REF: 011217a2 STA: A2.A.29 366 ANS: 31 − 19 12 = = 4 x + (4 − 1)4 = 19 a n = 7 + (n − 1)4 3 7−4 x + 12 = 19 TOP: Sequences 10 = x + 2(1) 8=x PTS: 2 365 ANS: 4 10 = 2.5 4 REF: 081014a2 x=7 011434a2 2 STA: A2.A.29 REF: 061520a2 TOP: Sequences STA: A2.A.29 2 REF: 061001a2 STA: A2.A.30 2 REF: 011110a2 STA: A2.A.30 REF: 011401a2 PTS: 2 STA: A2.A.30 REF: 061411a2 TOP: Sequences STA: A2.A.30 PTS: 2 373 ANS: 3 4 = −2 −2 REF: 081025a2 STA: A2.A.31 TOP: Sequences PTS: 2 REF: 011304a2 STA: A2.A.31 TOP: Sequences 367 368 369 370 PTS: 2 REF: ANS: 4 PTS: TOP: Sequences ANS: 3 PTS: TOP: Sequences ANS: 3 PTS: TOP: Sequences ANS: 1 (4a + 4) − (2a + 1) = 2a + 3 PTS: 2 371 ANS: 4 TOP: Sequences 372 ANS: 3 27r 4 − 1 = 64 r3 = 64 27 r= 4 3 16 ID: A 374 ANS: 2 3 − a3b4 32 6b = − 2 1 5 3 a a b 64 PTS: 2 375 ANS: 1 3 4 3 = − 2 1 − 2 REF: 061326a2 STA: A2.A.31 TOP: Sequences PTS: 2 376 ANS: 3 a n = 5(−2) n − 1 REF: 011508a2 STA: A2.A.31 TOP: Sequences STA: A2.A.32 TOP: Sequences a 15 = 5(−2) 15 − 1 = 81,920 PTS: 2 377 ANS: 1 a n = − 5(− REF: 011105a2 2) n − 1 a 15 = − 5(− 2) 15 − 1 = − 5(− 2 ) 14 = − 5 ⋅ 2 7 = −128 5 PTS: 2 REF: 061109a2 378 ANS: 3 40 − 10 30 = = 6 a n = 6n + 4 5 6−1 a 20 = 6(20) + 4 = 124 STA: A2.A.32 TOP: Sequences PTS: 2 379 ANS: −3,− 5,− 8,− 12 STA: A2.A.32 TOP: Sequences REF: 081510a2 PTS: 2 REF: fall0934a2 STA: A2.A.33 380 ANS: a 1 = 3. a 2 = 2(3) − 1 = 5. a 3 = 2(5) − 1 = 9. TOP: Recursive Sequences PTS: 2 381 ANS: TOP: Recursive Sequences a 2 = 3 2 PTS: 4 REF: 061233a2 −2 = STA: A2.A.33 3 −2 16 16 −2 27 3 a 3 = 3 = a 4 = 3 = 4 3 3 256 4 REF: 011537a2 STA: A2.A.33 17 TOP: Recursive Sequences ID: A 382 ANS: 3 3 3 8x 2x a 4 = 3xy 5 = 3xy 5 3 = 24x 4 y 2 y y PTS: 2 REF: 061512a2 STA: A2.A.33 383 ANS: 1 PTS: 2 REF: 081520a2 TOP: Sequences 384 ANS: 3 Σ n 0 1 2 2 n 2 0 2 2 2 2 12 n +2 0 +2 = 1 1 +2 = 3 2 +2 = 8 TOP: Sequences STA: A2.A.33 2 × 12 = 24 PTS: 2 REF: fall0911a2 STA: A2.N.10 TOP: Sigma Notation KEY: basic 385 ANS: 230. 10 + (1 3 − 1) + (2 3 − 1) + (3 3 − 1) + (4 3 − 1) + (5 3 − 1) = 10 + 0 + 7 + 26 + 63 + 124 = 230 PTS: 2 REF: 011131a2 STA: A2.N.10 KEY: basic 386 ANS: 1 Σ n 3 4 5 2 2 2 2 −r + r −3 + 3 = −6 −4 + 4 = −12−5 + 5 = −20 −38 TOP: Sigma Notation PTS: 2 KEY: basic 387 ANS: STA: A2.N.10 TOP: Sigma Notation STA: A2.N.10 TOP: Sigma Notation STA: A2.N.10 TOP: Sigma Notation REF: 061118a2 −104. PTS: 2 REF: 011230a2 KEY: basic 388 ANS: 4 4 + 3(2 − x) + 3(3 − x) + 3(4 − x) + 3(5 − x) 4 + 6 − 3x + 9 − 3x + 12 − 3x + 15 − 3x 46 − 12x PTS: 2 KEY: advanced REF: 061315a2 18 ID: A 389 ANS: 4 (a − 1) 2 + (a − 2) 2 + (a − 3) 2 + (a − 4) 2 (a 2 − 2a + 1) + (a 2 − 4a + 4) + (a 2 − 6a + 9) + (a 2 − 8a + 16) 4a 2 − 20a + 30 PTS: 2 REF: 011414a2 STA: A2.N.10 TOP: Sigma Notation KEY: advanced 390 ANS: 4 (3 − 2a) 0 + (3 − 2a) 1 + (3 − 2a) 2 = 1 + 3 − 2a + 9 − 12a + 4a 2 = 4a 2 − 14a + 13 PTS: 2 REF: 061526a2 KEY: advanced 391 ANS: x − 1 + x − 4 + x − 9 + x − 16 = 4x − 30 STA: A2.N.10 TOP: Sigma Notation PTS: KEY: 392 ANS: TOP: 393 ANS: STA: A2.N.10 TOP: Sigma Notation REF: 061025a2 STA: A2.A.34 2 REF: 081535a2 advanced 1 PTS: 2 Sigma Notation 15 ∑ 7n n=1 394 395 396 397 PTS: 2 REF: 081029a2 STA: A2.A.34 ANS: 2 PTS: 2 REF: 061205a2 TOP: Sigma Notation ANS: 1 PTS: 2 REF: 061420a2 TOP: Sigma Notation ANS: 4 PTS: 2 REF: 011504a2 TOP: Sigma Notation ANS: 4 n 21 S n = [2a + (n − 1)d] = [2(18) + (21 − 1)2] = 798 2 2 PTS: 2 REF: 061103a2 STA: A2.A.35 KEY: arithmetic 398 ANS: 3 n 19 S n = [2a + (n − 1)d] = [2(3) + (19 − 1)7] = 1254 2 2 PTS: 2 KEY: arithmetic REF: 011202a2 STA: A2.A.35 19 TOP: Sigma Notation STA: A2.A.34 STA: A2.A.34 STA: A2.A.34 TOP: Series TOP: Summations ID: A 399 ANS: . Sn = a n = 9n − 4 20(5 + 176) = 1810 2 a 1 = 9(1) − 4 = 5 a 20 = 9(20) − 4 = 176 PTS: 2 REF: 011328a2 KEY: arithmetic 400 ANS: 3 3(1 − (−4) 8 ) 196,605 S8 = = = −39,321 1 − (−4) 5 STA: A2.A.35 TOP: Summations PTS: 2 KEY: geometric 401 ANS: 1 STA: A2.A.35 TOP: Summations STA: A2.A.55 REF: 081010a2 TOP: Trigonometric Ratios STA: A2.A.55 cos K = REF: 061304a2 5 6 K = cos −1 5 6 K ≈ 33°33' PTS: 402 ANS: TOP: 403 ANS: 2 REF: 061023a2 2 PTS: 2 Trigonometric Ratios 1 12 2 − 6 2 = 108 = 36 3 = 6 3 . cot J = PTS: 2 REF: 011120a2 404 ANS: 2 PTS: 2 TOP: Trigonometric Ratios A 6 = ⋅ O 6 3 STA: A2.A.55 REF: 011315a2 20 3 3 = 3 3 TOP: Trigonometric Ratios STA: A2.A.55 ID: A 405 ANS: 2 sin S = 8 17 S = sin −1 8 17 S ≈ 28°4' PTS: 2 REF: 061311a2 406 ANS: 3 PTS: 2 TOP: Trigonometric Ratios 407 ANS: 3 5 10π 5π 2π ⋅ = = 12 12 6 STA: A2.A.55 REF: 061514a2 TOP: Trigonometric Ratios STA: A2.A.55 PTS: 2 REF: 061125a2 408 ANS: 2 PTS: 2 TOP: Radian Measure 409 ANS: STA: A2.M.1 REF: 061502a2 TOP: Radian Measure STA: A2.M.1 197º40’. 3.45 × 180 π ≈ 197°40 ′. PTS: 2 KEY: degrees 410 ANS: 2 11π 180 ⋅ = 165 12 π REF: fall0931a2 STA: A2.M.2 TOP: Radian Measure PTS: 2 KEY: degrees 411 ANS: 1 π = − 7π −420 3 180 REF: 061002a2 STA: A2.M.2 TOP: Radian Measure PTS: 2 KEY: radians REF: 081002a2 STA: A2.M.2 TOP: Radian Measure 21 ID: A 412 ANS: 180 2.5 ⋅ ≈ 143.2° π PTS: 2 KEY: degrees 413 ANS: 1 180 360 2⋅ = REF: 011129a2 STA: A2.M.2 TOP: Radian Measure PTS: 2 KEY: degrees 414 ANS: π ≈ 3.8 216 180 REF: 011220a2 STA: A2.M.2 TOP: Radian Measure PTS: 2 KEY: radians 415 ANS: REF: 061232a2 STA: A2.M.2 TOP: Radian Measure π 3× 180 π π ≈ 171.89° ≈ 171°53′. PTS: 2 KEY: degrees 416 ANS: 2 8π 180 ⋅ = 288 5 π REF: 011335a2 STA: A2.M.2 TOP: Radian Measure PTS: 2 KEY: degrees 417 ANS: 1 180 5⋅ ≈ 286 REF: 061302a2 STA: A2.M.2 TOP: Radian Measure PTS: 2 KEY: degrees REF: 011427a2 STA: A2.M.2 TOP: Radian Measure π 22 ID: A 418 ANS: 180 2.5 ⋅ ≈ 143°14' π PTS: 2 REF: 061431a2 KEY: degrees 419 ANS: 5 180 = 81°49' π 11 π STA: A2.M.2 TOP: Radian Measure PTS: 2 REF: 011531a2 KEY: degrees 420 ANS: 180 = 143°14' 2.5 π STA: A2.M.2 TOP: Radian Measure PTS: 2 KEY: degrees 421 ANS: STA: A2.M.2 TOP: Radian Measure REF: 081528a2 − 3 2 PTS: 2 REF: 061033a2 STA: A2.A.60 422 ANS: 4 PTS: 2 REF: 081005a2 TOP: Unit Circle 423 ANS: 4 PTS: 2 REF: 061206a2 TOP: Unit Circle 424 ANS: 3 If csc P > 0, sinP > 0. If cot P < 0 and sinP > 0, cos P < 0 TOP: Unit Circle STA: A2.A.60 PTS: 425 ANS: TOP: 426 ANS: TOP: TOP: Finding the Terminal Side of an Angle STA: A2.A.60 2 REF: 061320a2 STA: A2.A.60 3 PTS: 2 REF: 061412a2 Finding the Terminal Side of an Angle 4 PTS: 1 REF: 011312a2 Determining Trigonometric Functions 23 STA: A2.A.60 STA: A2.A.56 KEY: degrees, common angles ID: A 427 ANS: 2 6 3 × = 2 2 4 PTS: 2 REF: 061331a2 STA: A2.A.56 KEY: degrees, common angles 428 ANS: y 13 2 2 = = . sin θ = . csc θ = 2 2 2 2 2 13 x +y (−3) + 2 PTS: 2 429 ANS: 2 REF: fall0933a2 x2 + y2 = x sec θ = PTS: 2 430 ANS: 2 x = 2⋅ 3 = 2 STA: A2.A.62 TOP: Determining Trigonometric Functions 13 . 2 TOP: Determining Trigonometric Functions (−4) 2 + 0 2 4 = = −1 −4 −4 REF: 011520a2 3 y = 2⋅ STA: A2.A.62 TOP: Determining Trigonometric Functions 1 =1 2 PTS: 2 431 ANS: 2 REF: 061525a2 STA: A2.A.62 TOP: Determining Trigonometric Functions PTS: 2 432 ANS: 1 REF: 061115a2 STA: A2.A.66 TOP: Determining Trigonometric Functions PTS: 2 433 ANS: 4 REF: 011203a2 STA: A2.A.66 TOP: Determining Trigonometric Functions PTS: 2 REF: 061217a2 STA: A2.A.66 TOP: Determining Trigonometric Functions 24 ID: A 434 ANS: TOP: 435 ANS: TOP: 436 ANS: TOP: 437 ANS: 3 PTS: 2 REF: 081007a2 Using Inverse Trigonometric Functions 3 PTS: 2 REF: 011104a2 Using Inverse Trigonometric Functions 1 PTS: 2 REF: 011112a2 Using Inverse Trigonometric Functions 2 tan30 = STA: KEY: STA: KEY: STA: KEY: A2.A.64 basic A2.A.64 unit circle A2.A.64 advanced 3 3 = 30 . Arc cos 3 k 3 k = cos 30 k =2 PTS: 2 KEY: advanced 438 ANS: 2 REF: 061323a2 STA: A2.A.64 TOP: Using Inverse Trigonometric Functions 7 25 7 24 sin A 7 = = − If sin A = − , cos A = , and tanA = cos A 24 24 25 25 25 − PTS: 2 KEY: advanced 439 ANS: 1 REF: 011413a2 STA: A2.A.64 TOP: Using Inverse Trigonometric Functions 8 17 8 15 8 = If sin θ = , then cos θ = . tan θ = 15 15 17 17 17 PTS: 2 REF: 081508a2 KEY: advanced 440 ANS: 2 cos(−305° + 360°) = cos(55°) STA: A2.A.64 TOP: Using Inverse Trigonometric Functions PTS: 2 REF: 061104a2 441 ANS: 2 PTS: 2 TOP: Reference Angles 442 ANS: 4 s = θ r = 2⋅4 = 8 STA: A2.A.57 REF: 081515a2 TOP: Reference Angles STA: A2.A.57 STA: A2.A.61 TOP: Arc Length PTS: 2 KEY: arc length REF: fall0922a2 25 ID: A 443 ANS: 3 s=θr= 2π 3π ⋅6 = 8 2 PTS: 2 KEY: arc length 444 ANS: 83°50'⋅ π 180 REF: 061212a2 STA: A2.A.61 TOP: Arc Length ≈ 1.463 radians s = θ r = 1.463 ⋅ 12 ≈ 17.6 PTS: 2 REF: 011435a2 KEY: arc length 445 ANS: 2 2π s=θr= ⋅ 18 ≈ 23 5 STA: A2.A.61 TOP: Arc Length PTS: 2 KEY: arc length 446 ANS: 6.6 r= = 9.9 2 3 STA: A2.A.61 TOP: Arc Length PTS: 2 REF: 081532a2 STA: A2.A.61 KEY: radius 447 ANS: 3 Cofunctions tangent and cotangent are complementary TOP: Arc Length PTS: 2 REF: 061014a2 448 ANS: 3 1 sin 2 θ + cos 2 θ = = sec 2 θ 2 2 1 − sin θ cos θ STA: A2.A.58 TOP: Cofunction Trigonometric Relationships PTS: 2 449 ANS: STA: A2.A.58 TOP: Reciprocal Trigonometric Relationships REF: 061230a2 STA: A2.A.58 TOP: Reciprocal Trigonometric Relationships REF: 011330a2 STA: A2.A.58 TOP: Cofunction Trigonometric Relationships cos θ ⋅ REF: 011526a2 REF: 061123a2 1 − cos 2 θ = 1 − cos 2 θ = sin 2 θ cos θ PTS: 2 450 ANS: a + 15 + 2a = 90 3a + 15 = 90 3a = 75 a = 25 PTS: 2 26 ID: A 451 ANS: cos x sinx sin x cot x sinx = = cos 2 x sec x 1 cos x PTS: 2 452 ANS: 2 REF: 061334a2 STA: A2.A.58 TOP: Reciprocal Trigonometric Relationships cos x sinx cot x = = cos x csc x 1 sinx PTS: 2 REF: 061410a2 STA: A2.A.58 TOP: Reciprocal Trigonometric Relationships 453 ANS: 3 sin 2 x 1 cos 2 x 2 2 2 2 = sin 2 x − cos 2 x ≠ 1 = sin sin 2 x 1 + x + cos x = 1 cos x − 1 (cos x) = 1 2 2 cos 2 x sin x cos x 2 cos 2 x 1 x 1 cos − 1 = − = csc 2 x − cot x = 1 2 2 x x sin 2 x cos 2 x sin sin PTS: 2 REF: 011515a2 454 ANS: 1 −1 cos 2 x cos 2 x 1 − cos 2 x ⋅ = = sin 2 x 2 1 1 cos x cos 2 x STA: A2.A.58 TOP: Reciprocal Trigonometric Relationships PTS: 2 455 ANS: STA: A2.A.58 TOP: Reciprocal Trigonometric Relationships REF: 081533a2 3 3 2 3 2 ⋅ . If sin 60 = , then csc 60 = 3 2 3 PTS: 2 456 ANS: 1 sin 120 = REF: 011235a2 3 2 csc 120 = ⋅ 2 3 3 3 = 3 = 2 3 3 STA: A2.A.59 TOP: Reciprocal Trigonometric Relationships 2 3 3 PTS: 2 REF: 081505a2 STA: A2.A.59 457 ANS: 2 PTS: 2 REF: 011208a2 TOP: Simplifying Trigonometric Expressions 27 TOP: Reciprocal Trigonometric Relationships STA: A2.A.67 ID: A 458 ANS: sin 2 A cos 2 A 1 + = 2 2 cos A cos A cos 2 A tan2 A + 1 = sec 2 A PTS: 2 459 ANS: REF: 011135a2 sec θ sin θ cot θ = STA: A2.A.67 TOP: Proving Trigonometric Identities 1 cos θ ⋅ sin θ ⋅ =1 cos θ sin θ PTS: 2 REF: 011428a2 STA: A2.A.67 TOP: Proving Trigonometric Identities 460 ANS: 3 PTS: 2 REF: fall0910a2 STA: A2.A.76 TOP: Angle Sum and Difference Identities KEY: simplifying 461 ANS: 5 2 5 8 + 15 23 + 41 3 12 12 23 4 sinB 5 23 cos 2 B + sin 2 B = 1 tanB = = = tan(A + B) = = = = 2 4 12 10 2 cos B 4 2 2 5 − 1 − 5 2 12 12 12 41 3 4 cos 2 B + = 1 41 cos 2 B + 25 41 = 41 41 cos 2 B = cos B = 16 41 4 41 PTS: 4 REF: 081037a2 KEY: evaluating 462 ANS: sin(45 + 30) = sin 45 cos 30 + cos 45 sin 30 = STA: A2.A.76 6+ 4 3 2 1 6 2 2 ⋅ + ⋅ = + = 2 2 2 2 4 4 PTS: 4 REF: 061136a2 STA: A2.A.76 KEY: evaluating 463 ANS: 1 5 3 12 4 15 48 33 cos(A − B) = − + = − + = 5 13 5 65 65 65 13 PTS: 2 KEY: evaluating REF: 011214a2 STA: A2.A.76 28 TOP: Angle Sum and Difference Identities 2 TOP: Angle Sum and Difference Identities TOP: Angle Sum and Difference Identities ID: A 464 ANS: 1 sin(180 + x) = (sin 180)(cos x) + (cos 180)(sin x) = 0 + (−sinx) = −sinx PTS: 2 REF: 011318a2 STA: A2.A.76 TOP: Angle Sum and Difference Identities KEY: identities 465 ANS: 4 sin(θ + 90) = sin θ ⋅ cos 90 + cos θ ⋅ sin90 = sin θ ⋅ (0) + cos θ ⋅ (1) = cos θ PTS: 2 REF: 061309a2 KEY: identities 466 ANS: 2 cos(x − y) = cos x cos y + sin x siny STA: A2.A.76 TOP: Angle Sum and Difference Identities PTS: 2 REF: 061421a2 STA: A2.A.76 KEY: simplifying 467 ANS: 1 cos 2 θ − cos 2θ = cos 2 θ − (cos 2 θ − sin 2 θ) = sin 2 θ TOP: Angle Sum and Difference Identities PTS: 2 KEY: simplifying 468 ANS: 3 2 2 + cos 2 A = 1 3 TOP: Double Angle Identities = b ⋅b +a ⋅a = b2 + a2 REF: 061024a2 sin 2A = 2 sinA cos A 2 5 = 2 3 3 5 cos A = 9 2 cos A = + 5 , sin A is acute. 3 PTS: 2 KEY: evaluating 469 ANS: 1 STA: A2.A.77 = 4 5 9 REF: 011107a2 1 If sin x = 0.8, then cos x = 0.6. tan x = 2 PTS: 2 470 ANS: 4 REF: 061220a2 STA: A2.A.77 1 − 0.6 = 1 + 0.6 TOP: Double Angle Identities 0.4 = 0.5. 1.6 STA: A2.A.77 TOP: Half Angle Identities 2 1 2 7 cos 2A = 1 − 2 sin2 A = 1 − 2 = 1 − = 3 9 9 PTS: 2 KEY: evaluating REF: 011311a2 STA: A2.A.77 29 TOP: Double Angle Identities ID: A 471 ANS: 3 2 3 32 9 23 cos 2A = 1 − 2 sin 2 A = 1 − 2 = − = 8 32 32 32 PTS: 2 REF: 011510a2 STA: A2.A.77 KEY: evaluating 472 ANS: 1 1 + cos 2A 1 + 2 cos 2 A − 1 cos A = = = cot A sinA sin 2A 2 sinA cos A TOP: Double Angle Identities PTS: 2 REF: 061522a2 STA: A2.A.77 KEY: simplifying 473 ANS: 1 2 3 9 9 8 1 cos 2θ = 2 − 1 = 2 − 1 = − = 8 8 8 4 16 TOP: Double Angle Identities PTS: 2 KEY: evaluating 474 ANS: 1 TOP: Double Angle Identities tan θ − REF: 081522a2 3 =0 tan θ = . STA: A2.A.77 . 3 θ = tan−1 3 θ = 60, 240 PTS: 2 KEY: basic REF: fall0903a2 STA: A2.A.68 30 TOP: Trigonometric Equations ID: A 475 ANS: 0, 60, 180, 300. sin 2θ = sin θ sin 2θ − sin θ = 0 2 sin θ cos θ − sin θ = 0 sin θ(2 cos θ − 1) = 0 sin θ = 0 2 cos θ − 1 = 0 θ = 0,180 cos θ = 1 2 θ = 60,300 PTS: 4 REF: 061037a2 KEY: double angle identities 476 ANS: 45, 225 2 tan C − 3 = 3 tan C − 4 STA: A2.A.68 TOP: Trigonometric Equations STA: A2.A.68 TOP: Trigonometric Equations STA: A2.A.68 TOP: Trigonometric Equations 1 = tanC tan−1 1 = C C = 45,225 PTS: 2 KEY: basic 477 ANS: 4 REF: 081032a2 2 cos θ = 1 cos θ = 1 2 θ = cos −1 PTS: 2 KEY: basic 1 = 60, 300 2 REF: 061203a2 31 ID: A 478 ANS: 3 − 2 sec x = 2 sec x = − 2 2 cos x = − 2 2 x = 135,225 PTS: 2 REF: 011322a2 KEY: reciprocal functions 479 ANS: 5 csc θ = 8 STA: A2.A.68 TOP: Trigonometric Equations STA: A2.A.68 TOP: Trigonometric Equations REF: 011436a2 STA: A2.A.68 TOP: Trigonometric Equations PTS: 2 REF: 061434a2 KEY: reciprocal functions STA: A2.A.68 TOP: Trigonometric Equations csc θ = 8 5 sin θ = 5 8 θ ≈ 141 PTS: 2 REF: 061332a2 KEY: reciprocal functions 480 ANS: 2 sin 2 x + 5 sin x − 3 = 0 (2 sinx − 1)(sinx + 3) = 0 sinx = x= PTS: 4 KEY: quadratics 481 ANS: sec x = 2 cos x = 1 2 cos x = 2 2 1 2 π 5π 6 , 6 x = 45°,315° 32 ID: A 482 ANS: 5 cos θ − 2 sec θ + 3 = 0 5 cos θ − 2 +3= 0 cos θ 5 cos 2 θ + 3 cos θ − 2 = 0 (5 cos θ − 2)(cos θ + 1) = 0 cos θ = 2 ,−1 5 θ ≈ 66.4,293.6,180 PTS: 6 REF: 061539a2 KEY: reciprocal functions 483 ANS: 2 (2 sinx − 1)(sinx + 1) = 0 sin x = STA: A2.A.68 TOP: Trigonometric Equations STA: A2.A.68 TOP: Trigonometric Equations 1 ,−1 2 x = 30,150,270 PTS: KEY: 484 ANS: 2π = b 2 quadratics 4 2π = 6π 1 3 REF: 081514a2 PTS: TOP: 485 ANS: 2π = b 2 REF: 061027a2 STA: A2.A.69 Properties of Graphs of Trigonometric Functions 2 2π 3 PTS: 2 REF: 061111a2 STA: A2.A.69 TOP: Properties of Graphs of Trigonometric Functions 486 ANS: 1 2π = 4π b b= KEY: period KEY: period 1 2 PTS: 2 REF: 011425a2 STA: A2.A.69 TOP: Properties of Graphs of Trigonometric Functions 33 KEY: period ID: A 487 ANS: 2 2π π = 6 3 PTS: 2 REF: 061413a2 STA: A2.A.69 TOP: Properties of Graphs of Trigonometric Functions 488 ANS: 1 2π =π 2 KEY: period π =1 π PTS: 2 REF: 061519a2 STA: A2.A.69 TOP: Properties of Graphs of Trigonometric Functions 489 ANS: 3 2π =π 2 PTS: 2 REF: 081519a2 STA: A2.A.69 TOP: Properties of Graphs of Trigonometric Functions 490 ANS: 4 2π = 30 b b= KEY: period KEY: period π 15 PTS: 2 REF: 011227a2 STA: A2.A.72 TOP: Identifying the Equation of a Trigonometric Graph 491 ANS: y = −3sin 2x . The period of the function is π , the amplitude is 3 and it is reflected over the x-axis. PTS: 2 REF: 061235a2 STA: A2.A.72 TOP: Identifying the Equation of a Trigonometric Graph 492 ANS: 1 PTS: 2 REF: 011320a2 TOP: Identifying the Equation of a Trigonometric Graph 493 ANS: 3 PTS: 2 REF: 061306a2 TOP: Identifying the Equation of a Trigonometric Graph 494 ANS: a = 3, b = 2, c = 1 y = 3cos 2x + 1. PTS: TOP: 495 ANS: TOP: 496 ANS: TOP: 2 REF: 011538a2 STA: A2.A.72 Identifying the Equation of a Trigonometric Graph 3 PTS: 2 REF: fall0913a2 Graphing Trigonometric Functions 3 PTS: 2 REF: 061119a2 Graphing Trigonometric Functions 34 STA: A2.A.72 STA: A2.A.72 STA: A2.A.65 STA: A2.A.65 ID: A Algebra 2/Trigonometry Regents Exam Questions by Performance Indicator: Topic Answer Section 497 ANS: 3 period = 2π 2π 2 = = 3 b 3π PTS: 2 KEY: recognize 498 ANS: 3 REF: 081026a2 STA: A2.A.70 TOP: Graphing Trigonometric Functions PTS: 2 499 ANS: 1 REF: 061020a2 STA: A2.A.71 TOP: Graphing Trigonometric Functions PTS: 2 500 ANS: 3 REF: 011123a2 STA: A2.A.71 TOP: Graphing Trigonometric Functions 011207a2 2 STA: A2.A.71 REF: 061022a2 TOP: Graphing Trigonometric Functions STA: A2.A.63 2 REF: 061224a2 STA: A2.A.63 2 REF: 061427a2 STA: A2.A.63 STA: A2.A.74 TOP: Using Trigonometry to Find Area 501 502 503 504 PTS: 2 REF: ANS: 3 PTS: TOP: Domain and Range ANS: 3 PTS: TOP: Domain and Range ANS: 4 PTS: TOP: Domain and Range ANS: 2 1 K = (10)(18) sin 120 = 45 2 PTS: 2 KEY: basic 3 ≈ 78 REF: fall0907a2 1 ID: A 505 ANS: K = absinC = 24 ⋅ 30 sin57 ≈ 604 PTS: 2 REF: 061034a2 KEY: parallelograms 506 ANS: 3 K = (10)(18) sin46 ≈ 129 STA: A2.A.74 TOP: Using Trigonometry to Find Area PTS: 2 REF: 081021a2 KEY: parallelograms 507 ANS: 1 1 (7.4)(3.8) sin 126 ≈ 11.4 2 STA: A2.A.74 TOP: Using Trigonometry to Find Area PTS: 2 KEY: basic 508 ANS: STA: A2.A.74 TOP: Using Trigonometry to Find Area REF: 011218a2 K = absinC = 18 ⋅ 22sin60 = 396 3 = 198 3 2 PTS: 2 REF: 061234a2 KEY: Parallelograms 509 ANS: 3 1 42 = (a)(8) sin 61 2 STA: A2.A.74 TOP: Using Trigonometry to Find Area STA: A2.A.74 TOP: Using Trigonometry to Find Area PTS: 4 REF: 061337a2 KEY: advanced 511 ANS: K = absinC = 6 ⋅ 6 sin50 ≈ 27.6 STA: A2.A.74 TOP: Using Trigonometry to Find Area PTS: 2 REF: 011429a2 KEY: Parallelograms STA: A2.A.74 TOP: Using Trigonometry to Find Area 42 ≈ 3.5a 12 ≈ a PTS: 2 REF: 011316a2 KEY: basic 510 ANS: a 1 15 . (15)(10.3) sin35 ≈ 44 = sin 103 sin 42 2 a ≈ 10.3 2 ID: A 512 ANS: 2 1 (22)(13) sin 55 ≈ 117 2 PTS: 2 KEY: basic 513 ANS: 594 = 32 ⋅ 46 sinC REF: 061403a2 STA: A2.A.74 TOP: Using Trigonometry to Find Area PTS: 2 REF: 011535a2 KEY: Parallelograms 514 ANS: 2 STA: A2.A.74 TOP: Using Trigonometry to Find Area 594 = sinC 1472 23.8 ≈ C K = 8 ⋅ 12 sin120 = 96 ⋅ 3 = 48 3 2 PTS: 2 REF: 061508a2 STA: A2.A.74 TOP: Using Trigonometry to Find Area KEY: parallelograms 515 ANS: 10 c 12 12 = = . C ≈ 180 − (32 + 26.2) ≈ 121.8. sin 32 sinB sin32 sin 121.8 B = sin −1 10 sin32 ≈ 26.2 12 PTS: 4 REF: 011137a2 KEY: basic 516 ANS: x T 100 = . sin66 ≈ 88. 97.3 sin 33 sin 32 x ≈ 97.3 PTS: KEY: 518 ANS: TOP: 12 sin121.8 ≈ 19.2 sin 32 STA: A2.A.73 TOP: Law of Sines STA: A2.A.73 TOP: Law of Sines STA: A2.A.73 TOP: Law of Sines REF: 061322a2 KEY: modeling STA: A2.A.73 t ≈ 88 PTS: 4 REF: 011236a2 KEY: advanced 517 ANS: b a 100 100 = = . sin 32 sin 105 sin 32 sin 43 b ≈ 182.3 c= a ≈ 128.7 4 REF: 011338a2 basic 2 PTS: 2 Law of Sines 3 ID: A 519 ANS: 3 59.2 60.3 = 180 − 78.3 = 101.7 sin74 sinC C ≈ 78.3 PTS: 2 REF: 081006a2 520 ANS: 2 10 13 = . 35 + 48 < 180 sin 35 sinB 35 + 132 < 180 B ≈ 48,132 STA: A2.A.75 TOP: Law of Sines - The Ambiguous Case PTS: 2 REF: 011113a2 STA: A2.A.75 TOP: Law of Sines - The Ambiguous Case 521 ANS: 1 9 10 = . 58° + 70° is possible. 122° + 70° is not possible. sinA sin 70 A ≈ 58 PTS: 2 522 ANS: 1 REF: 011210a2 STA: A2.A.75 TOP: Law of Sines - The Ambiguous Case PTS: 2 REF: 061226a2 STA: A2.A.75 523 ANS: 4 13 20 . 81 + 40 < 180. (180 − 81) + 40 < 180 = sin40 sinM TOP: Law of Sines - The Ambiguous Case 6 10 = sin 35 sin N N ≈ 73 73 + 35 < 180 (180 − 73) + 35 < 180 M ≈ 81 PTS: 2 524 ANS: 2 8 5 = sin 32 sinE E ≈ 57.98 PTS: 2 REF: 061327a2 STA: A2.A.75 TOP: Law of Sines - The Ambiguous Case STA: A2.A.75 TOP: Law of Sines - The Ambiguous Case 57.98 + 32 < 180 (180 − 57.98) + 32 < 180 REF: 011419a2 4 ID: A 525 ANS: 4 34 12 = sin 30 sinB B = sin −1 ≈ sin −1 12 sin30 34 6 5.8 PTS: 2 REF: 011523a2 STA: A2.A.75 526 ANS: 2 8 85 + 14.4 < 180 1 triangle = sin 85 sinC 2 sin 85 85 + 165.6 ≥ 180 −1 C = sin 8 TOP: Law of Sines - The Ambiguous Case C ≈ 14.4 PTS: 2 527 ANS: 33. a = REF: 061529a2 STA: A2.A.75 TOP: Law of Sines - The Ambiguous Case 10 2 + 6 2 − 2(10)(6) cos 80 ≈ 10.7. ∠C is opposite the shortest side. 10.7 6 = sinC sin80 C ≈ 33 PTS: 6 REF: 061039a2 KEY: advanced 528 ANS: 4 7 2 = 3 2 + 5 2 − 2(3)(5) cos A STA: A2.A.73 TOP: Law of Cosines STA: A2.A.73 TOP: Law of Cosines 49 = 34 − 30 cos A 15 = −30 cos A 1 − = cos A 2 120 = A PTS: 2 REF: 081017a2 KEY: angle, without calculator 5 ID: A 529 ANS: 1 13 2 = 15 2 + 14 2 − 2(15)(14) cos C 169 = 421 − 420 cos C −252 = −420 cos C 252 = cos C 420 53 ≈ C PTS: 2 KEY: find angle 530 ANS: REF: 061110a2 STA: A2.A.73 TOP: Law of Cosines STA: A2.A.73 TOP: Law of Cosines 27 2 + 32 2 − 2(27)(32) cos 132 ≈ 54 PTS: KEY: 531 ANS: TOP: 532 ANS: 28 2 4 REF: 011438a2 applied 2 PTS: 2 Law of Cosines REF: 011501a2 STA: A2.A.73 KEY: side, without calculator = 47 2 + 34 2 − 2(47)(34) cos A 784 = 3365 − 3196 cos A −2581 = −3196 cos A 2581 = cos A 3196 36 ≈ A PTS: 4 KEY: find angle 533 ANS: a= REF: 061536a2 STA: A2.A.73 TOP: Law of Cosines 8 2 + 11 2 − 2(8)(11) cos 82 ≈ 12.67. The angle opposite the shortest side: 8 12.67 = sinx sin82 x ≈ 38.7 PTS: 4 KEY: advanced REF: 081536a2 STA: A2.A.73 6 TOP: Law of Cosines ID: A 534 ANS: r 2 = 25 2 + 85 2 − 2(25)(85) cos 125. 101.43, 12. r 2 ≈ 10287.7 r ≈ 101.43 101.43 2.5 = sinx sin 125 x ≈ 12 PTS: 6 535 ANS: REF: fall0939a2 STA: A2.A.73 TOP: Vectors STA: A2.A.73 TOP: Vectors F1 F2 27 27 = = . . sin 75 sin 60 sin 75 sin 45 F 1 ≈ 24 F 2 ≈ 20 PTS: 4 536 ANS: REF: 061238a2 R= 28 2 + 40 2 − 2(28)(40) cos 115 ≈ 58 40 58 = sin 115 sinx x ≈ 39 PTS: 6 REF: 061439a2 537 ANS: 2 x 2 − 2x + y 2 + 6y = −3 STA: A2.A.73 TOP: Vectors STA: A2.A.47 TOP: Equations of Circles x 2 − 2x + 1 + y 2 + 6y + 9 = −3 + 1 + 9 (x − 1) 2 + (y + 3) 2 = 7 PTS: 2 REF: 061016a2 7 ID: A 538 ANS: 3 x 2 + y 2 − 16x + 6y + 53 = 0 x 2 − 16x + 64 + y 2 + 6y + 9 = −53 + 64 + 9 (x − 8) 2 + (y + 3) 2 = 20 PTS: 2 539 ANS: 4 r= (6 − 3) 2 + (5 − (−4)) 2 = PTS: 2 540 ANS: 3 r= REF: 011415a2 9 + 81 = REF: 061415a2 (6 − 2) 2 + (2 − −3) 2 = 16 + 25 = STA: A2.A.47 TOP: Equations of Circles 90 STA: A2.A.48 TOP: Equations of Circles 41 PTS: 2 REF: 081516a2 541 ANS: (x + 3) 2 + (y − 4) 2 = 25 STA: A2.A.48 TOP: Equations of Circles PTS: 2 REF: fall0929a2 542 ANS: (x + 5) 2 + (y − 3) 2 = 32 STA: A2.A.49 TOP: Writing Equations of Circles PTS: 2 REF: 081033a2 543 ANS: 2 PTS: 2 TOP: Equations of Circles 544 ANS: STA: A2.A.49 REF: 011126a2 TOP: Writing Equations of Circles STA: A2.A.49 r= PTS: 545 ANS: TOP: 546 ANS: TOP: 22 + 32 = 13 . (x + 5) 2 + (y − 2) 2 = 13 2 REF: 011234a2 4 PTS: 2 Equations of Circles 4 PTS: 2 Equations of Circles STA: A2.A.49 REF: 061318a2 TOP: Writing Equations of Circles STA: A2.A.49 REF: 011513a2 STA: A2.A.49 8